So about a week or so ago I got a question (for those that I have not yet answered, I am getting around to them – really need to change how I get these questions!) which related to Islam.
Dear Rabbi, I am so happy that I can write to you. I am a Christian and have a great respect for your Faith.
I have always been one to believe people should not really need to convert just to marry someone and be with them but I am aware the Islāmic community is quite vocal on that point. Now so I do not confuse any readers, I am going to be using the name Allah for The G'd of the Qur'an and Yahweh for the G'd of the Jewish / Christian faith.
There is little doubt there is a connection between the Hebrew Bible and the Christian Bible and that they both worship and accept the same G'd – Yahweh although they both do so in different ways. The main different between the two beliefs is how they understand G'd which mostly goes down to the Christian view of the trinity and thus Yahweh is the G'd of Abraham – something no one really will debate.
However when it comes to Allah this is not so clear although a Muslim will tell you otherwise. The reason for this is unlike the origins of Judaism and by proxy, Christianity Islam came about independently from what I am going to call "Abrahamism". Islam itself claims they are descendants of Ishmael … but there is no proof of that and there is no evidence in the Tanakh (Hebrew Bible) or anywhere else basically (although the core of Islam does claim Jews edited the holy books etc).
Now Islam does claim to be a Abrahamic religion but the thing is nothing came from Abraham at all. With the Jewish faith everything has been passed down from generations to the next descendants while with Islam it was passed down to Mohammad by an angel (not sure which one, will research it later).
As you can see there is not really a connection between Islam and Judaism (although the religions are alike).
Now with all that in mind, its time to answer the question. There is little connection between Judaism and Islam (like there is with Christianity) but Islam claims they are the same G'd but just like with the Christian faith, they have a different way of understanding and accepting G'd.
So to answer your question in a few words – They worship the same G'd with a different understanding and holy text. With all that said the two religions believe in different things and although Judaism is "meant" to be respected within Islam, I have found that rarely to be the case.