You asked: "But broadly speaking, Christianity and Judaism seek to serve the same God, do they not?" I don't really think they do. What makes you say this? What are the elements of similarity? The Jewish and Christian conceptions of G-d are very very different. Jews fundamentally reject Jesus as Christ--we fundamentally reject the *idea* of a deity in human form. You asked "And is not their God the God of Abraham, Isaac, Moses, and Elijah?". The answer is that, yes, Jews and Christians believe in a link between their G-d and the G-d of the people you mentioned (Jews might traditionally refer to "the G-d of Avraham, Yitzhak v'Ya'akov" in this context). So, of course, do Muslims. And, of course, Muslims also believe in the 'Old Testament' as being sacred scripture. They differ from Jews in that they believe the same of the 'New Testament'; Judaism rejects the 'New Testament' entirely and *never* refers to the Torah as the 'Old Testament'. Judaism also traditionally believe in the Talmud as being sacred -- Torah she'be'al'peh (Torah that is in the mouth, as opposed to written). That adds a collection of more than a million words of Jewish thought that is unique and not at all Christian (or Islamic) in outlook.
I think it is easy both to over-play and to under-play the resemblance of each of the three religions to each other. They are distinct and distinctive, sister faiths. They share some important concepts -- and are utterly unlike each other in other ways.
The First Christian Church was in Jerusalem and consisted of Jews. The origination of Christianity was totally Jewish -- as there were absolutely no Gentiles in the beginning (but, of course, later on -- as evidenced by Peter under guidance of the Holy Spirit and from prophetic Scriptures of the Jews).
There was no such thing with Islam and they set themselves against Christians and Jews in their own writings.
Christians don't set themselves against Jews in their Holy Scriptures -- and in fact -- say that they are part of the same belief and same God. Jesus, Himself makes that claim. And Christians accept totally and 100% the Holy Written Scriptures -- the law, prophets and writings of the Jews, that Jesus accepted, too -- in their entirety.
There is *no such thing* with Islam - in any way.