Well, to reply to the original question:
IF Muslims don’t believe in the same God as Christians, do Jews?
Simple answer: If the first premise is true, then the answer is No. Jews do not believe in the same God as Christians.
And why not? Because Jews and Muslims readily admit that they believe in the same God. They sure express their worship in different ways, but go to a Jewish or Islamic board and have this conversation and you will find that the majority of both Jews and Muslim recognize that they worship the same God, though they use different names to address him.
However, the premise is false. All three, Christianity, Judaism, and Islam worship the God of Abraham. They claim to know different things about this God. They claim that the revelation they have received and hold to is correct and that the others either have not the truth in their revelation or interpret it falsely or both. But however differently they worship and however differently they believe, it is still the same God.
What about the objection that Muslims don’t believe in Jesus? Well, actually they do, they just don’t worship Jesus. Further, this same objection is the reason for our question with respect to the Jews. So, if Muslims don’t worship the same God as Christians because they don’t worship Jesus, then the same standard would eliminate Jews from worshipping the same God as Christians. This would put Christians in the strange state of saying that Jesus, who worshipped the same God as all others Jews, worshipped a different God than Christians.
As this latter makes no sense, then it follows that the standard which would discount Jews for failure to worship Jesus must not the the standard to determine whether a particular religion worships the same God as Chrisitianity or not. Thus, we cannot apply it to Muslims either.
We are back to three different religions claiming to all worship the God of Abraham, but claiming completely different things as being true about that God. Can this be recognized as worshipping the same God? I believe it can. Other than a few that want to ostracize everyone who thinks even slightly differently than they themselves do, most Christians recognize that even though they have different beliefs about God (such as whether the Holy Spirit proceeds from the Father alone or from the Father and the Son; or such as whether or not Christ is physically present in the host in the eucharist) that Catholic, Orthodox, and Protestant Christians all believe in the same God. Indeed, probably no two Christians believe exactly the same with respect to God (not even the first century Church if we are to believe the report of the Council in Jerusalem from Acts 15). And yet, we don’t hold the the notion that we each believe in a different God. So, it is possible to have different, even conflict beliefs and yet to believe in the same God – though I would propose that at least one is probably believing incorrectly. And thus it can be that all three, Christianity, Islam, and Judaism all believe in the same God – though of course, IMO, Christiantiy comes the closest to getting it right.