When is the age of accountability? This is an interesting question. Not interesting as in it's a fun topic to discuss, but interesting in the fact that, despite being illogical, it is the right question to ask.
In logic, there is a concept known as a "fallacy". That is, a failure in reasoning that renders an argument or statement illogical. One of those fallacies is called a "loaded question". A loaded question is a question that assumes an unestablished premise. The common example, "have you stopped beating your wife?" The question is loaded with the assumption that you ever beat your wife in the first place. If you say "yes", you beat your wife, but don't anymore. If you say "no", you are still beating your wife. Similarly, "when is the age of accountability" leaves no room for the possibility that there is no such thing as the age of accountability. Thus, the statement is illogical. But in spite of being illogical, it is actually a good question. I believe it betrays an innate knowledge of God's Holy character. God, as a God of love and justice, likely isn't going to hold people accountable for sins they cannot even commit.
Let me get deep for a moment. Let's suppose the age of accountability is a myth. Let's suppose it's a lovely thought, but is theologically unsustainable. Let us imagine (and I believe it would be imaginary) that God is capable of condemning even a child to Hell. This thought, though offensive, must nevertheless be "acceptable". As humans, we are both sinful and fallible. God, by contrast, is Holy and infallible. If it is just to condemn a child, we have to understand that God would know it, even if we don't, and that though we naturally find such a conclusion repugnant in every sense, we must still trust Him nevertheless. However, with such a detestable thought out of the way, I take great pleasure in saying that I believe a solid Biblical case can be made for the existence of something like the age of accountability. Allow me to make that case now.
The first hint of an age of accountability comes from one of Moses' books. In Deuteronomy 1:34-39, we read "And the Lord heard the sound of your words, and was angry, and took an oath, saying, ‘Surely not one of these men of this evil generation shall see that good land of which I swore to give to your fathers, except Caleb the son of Jephunneh; he shall see it, and to him and his children I am giving the land on which he walked, because he wholly followed the Lord.’ The Lord was also angry with me for your sakes, saying, ‘Even you shall not go in there. Joshua the son of Nun, who stands before you, he shall go in there. Encourage him, for he shall cause Israel to inherit it. ‘Moreover your little ones and your children, who you say will be victims, who today have no knowledge of good and evil, they shall go in there; to them I will give it, and they shall possess it."
From this, we already see God distinguishing between those who are old and accountable, and those "who today have no knowledge of good and evil", namely "your little ones and your children". This directly hints at a concept that is later explained in the New Testament, namely that knowledge is required for sin. Or, to put it as Paul does, "sin is not imputed where there is no law" (Romans 5:13). This would explain why, in 1 Corinthians 14:20, Paul exhorts us "Brethren, do not be like children in understanding; however, in malice, be babes, but in understanding be mature." There is no malice in a child. Children are notoriously sweet and innocent, even kind. They are almost naive. And so what have they to be accountable for? What charge can be brought against a being who can be controlled with a packet of jelly beans? What crime can be attributed to a wriggling ball of flailing limbs? A baby's neck cannot even support its head, and so how can they rebel against the Lord?
And we see from Christ's own words that accountability doesn't begin when a child can walk. Consider Matthew 18:1-5: "At that time the disciples came to Jesus, saying, “Who then is greatest in the kingdom of heaven?” Then Jesus called a little child to Him, set him in the midst of them, and said, “Assuredly, I say to you, unless you are converted and become as little children, you will by no means enter the kingdom of heaven. Therefore whoever humbles himself as this little child is the greatest in the kingdom of heaven. Whoever receives one little child like this in My name receives Me." So what do we see here? We see a child who can clearly walk, and even receive commands. Jesus sees the little child, calls the child, and the child obeys. Jesus' conclusion? Humble yourself as the little child, and you'll be the greatest in Heaven, whereas unless you become converted and act as little children, you cannot even get there.
Now, who humbles themselves, and becomes as little children, quite like actual little children? Perhaps this is why Jesus, upon getting annoyed that the disciples were stopping children from approaching Him, declared "of such is the Kingdom of Heaven". Jesus loved children, and used them as examples for how we are to respond to Him. How, then, could He possibly condemn them for sins they aren't even capable of committing?
And so clearly, there is something akin to the age of accountability. However, asking when it is, I believe, is futile. How can we possibly quantify it? If we take the approach of some and try to pin down the age, we do a great disservice to our faith. First, if such an age did exist, the Bible would not have left it so ambiguous. No doubt a book designed to thoroughly instruct us in our faith would warn us of a time when our children instantly transition from innocent babes to fully accountable sinners. But in truth, though I believe I have made the case that the age of accountability exists, the Bible does not even mention an age of accountability.
It does, however, mention an "age for love". In Ezekiel 16, specifically verses 6-8, we read "“And when I passed by you and saw you struggling in your own blood, I said to you in your blood, ‘Live!’ Yes, I said to you in your blood, ‘Live!’ I made you thrive like a plant in the field; and you grew, matured, and became very beautiful. Your breasts were formed, your hair grew, but you were naked and bare. “When I passed by you again and looked upon you, indeed your time was the time of love; so I spread My wing over you and covered your nakedness. Yes, I swore an oath to you and entered into a covenant with you, and you became Mine,” says the Lord God."
This is God's metaphorical way of describing His relationship with Israel. Note how He describes Israel as a growing girl. From a babe whom no one cared enough to swaddle up to a beautiful woman with long hair, and well-developed breasts, and hers was the "time of love" (some translations render it "age of love"). In other words, she was mature enough to marry and bear children. Or at least, to avoid being crude, physically mature enough to engage in the act of trying.
In civilised countries, there is what is called the "age of consent". A person must be legally a certain age in order to engage in sexual activity. Engaging in sexual activity with those below such an age results in prosecution. For example, here in the UK, the age of consent is 16. If a 16 year old sleeps with a 14 year old, the 16 year old can be prosecuted for statutory rape (I am unsure of exactly what happens if both partners are below the age of consent). The 14 year old is not considered old enough to consent, and so even if they do consent, it is considered rape.
Consent laws draw criticism from both sides. Some argue that they are too strict. After all, legally speaking, if someone is just one day away from their 16th birthday, and sleeps with someone above that age, the one above the age can still be prosecuted even though the minor will be exactly as mature, physically and mentally, the next day, when it becomes legal. By contrast, others argue that the age of consent may actually be too low. Think about it, do you consider 16 year olds mature enough to make that decision? In truth, your answer may vary. I would probably say no, and certainly think 18 would be a safer standard to set. However, evidently, the legislators who set it at 16 would disagree with me. But in the U.S., some states set it at 17, others at 18. In South Korea, it is 20. Bahrain has the highest age of consent in the world, setting it at 21. Why the disparity? Simple: There is no such thing as an age for love. Because people mature at different rates, some people will be both physically and mentally ready for marriage at 16, whereas others may not be ready until they are 21. Consent laws exist to protect the "late bloomers", not to say everyone becomes instantly ready for marriage at exactly the same age.
In the same way, trying to establish an age of accountability is asinine, because everyone matures at different rates. The age of accountability cannot be set, because some people will reach it earlier, and some people will reach it later. God alone knows exactly when to start holding people accountable. Which is lucky for us, because let's be honest, we'd mess that right up. But what we can say is that a person becomes accountable for their actions when they are capable of assessing the moral implications. The moment they know they have a choice between good and evil, they become accountable for choosing evil.
It's interesting to note that Jesus Himself seems to have gone through a moral development stage. Of course, regardless of His upbringing, there is just no way Jesus would have sinned. However, in the prophecy of His birth in Isaiah 7, specifically in verses 15-16, it's said that He would eat curds and honey so He would know to refuse evil and choose good. In other words, Jesus went through the exact same process as every other human being. He truly was fully human. But He was also fully God, and so whereas our sinful nature causes us to go astray as soon as we are able, Jesus never went astray. But there was a time in His life when He didn't even know the difference. Jesus experienced life as a normal baby, with as blank a mind as any other. Thus, even while we think of when children become accountable, we should think of the time when a child was born to give account for us. The sins we commit, this child was destined to take the punishment for. Isn't that just mind blowing?