It is often claimed that the three Abrahamic religions, Islam, Christianity, and Judaism, all worship the same God. This is demonstrably false. The Allah of the Qur'an is a radically different god than the God of the Bible, so much so that while Muslims do often try to twist the Bible to support their views, they will often resort to arguing that the Bible has been corrupted since Muhammad's time.
But what about the Jews? This is not as black and white. The God they claim to serve, after all, really did inspire their Scriptures. These are the very Scriptures we, ourselves, have. So are Jews not worshiping the same God?
There are two ways to answer this question. The first is to simply say no. The Bible makes it perfectly clear that without the Son, you don't have the Father. Jews lack the Son, therefore Jews lack the Father. Different God. But this might be a bit too wooden. After all, while they do not accept the fullness of the revelation, they nevertheless share our Old Testament Scriptures, and of course, before the New Testament, that was all even the most faithful of God's people had. Thus, it almost seems self destructive to say they don't have the same God.
Let's talk about Abraham for a moment. Abraham is famous for his faith. In fact, he is the very picture of how faith saves. Abraham believed God, and it was credited to him for righteousness. But what was he believing? Abraham believed God's very specific promise. "This one shall not be your heir, but one who will come from your own body shall be your heir.” Then He brought him outside and said, “Look now toward heaven, and count the stars if you are able to number them.” And He said to him, “So shall your descendants be.”" (Genesis 15:4b-5).
In hindsight, we know God was talking about Isaac. But at the time of this revelation, Abraham had no idea who Isaac was. He knew that he would have an heir, but at this point, that heir had not been conceived. Thus, it was easy enough to believe God. But as time went on, that promise was fulfilled. Abraham's wife, Sarah, bore her the promised son, Isaac. Abraham's understanding of the promise was now increased, ready for him to reassess.
Now imagine if Abraham didn't like Isaac. "I don't like this kid, I'll wait for God to give me that heir He promised." If this had happened, Abraham's initial acceptance of the very same child would have become acceptance of a brand new fabrication.
Before Christ, the Jews expected the promised Messiah. They had a number of expectations about Him, but because they were all centered around the promise of God, it could be legitimately said that they had the same God. They just hadn't yet received His full revelation. But when He came, they rejected Him. Modern Jews still wait for the promised Messiah, though He has long been with us. Thus, the Jewish expectation of the Messiah switched from a faithful reliance on God's promise to a rejection thereof.
One might think of the Old Testament as a shadow. A shadow is not always clear. It tells us something is blocking the light, but it doesn't show all the detail of exactly what that is. But then, the thing casting the shadow comes into view. Where once the Lord's shadow was all the faithful had, He is now fully in view. The problem the Jews have is that, save a faithful remnant of them who do know Jesus as Lord, the majority of them continue to dance in the shadow. Thus, they have the same God, but they do not have the same God.