Granted the PERSPECTIVE of God held by Muslims and Christians and Jews is different, but because their PERSPECTIVE of God is different from ours, does not mean they recognise another God.
According to Muslim belief, both Christians and Jews are "people of the book", as they follow the old Bible upon which Islam is in part based. Accordingly, Christians and Jews are not to be stopped from practising their faith or forcefully converted. This is a distinction Muslims make between the latter and pagans who worship idols. Hence Muslims themselves feel Christians and Jews worship the same God Muslims worship. In a sense they view Christianity and Judaism as an unevolved arrested faith in the same sense that Christians tend to view Judaism or Protestants tned to view Catholicism.
The entire subject of Islam and Christianity and Judaism is a difficult one. Just as Christianity comes in different sects, so does Islam. While there are certain elements of Islam that westerners find difficult to accept, there are certain other asapects of their faith that are commendable - their modesty, their dedidication to charity, to devotion to a monotheistic God, etc. I must confess that I have an ambivalent mind on this subject and really can't decide how to view these people or their religion.
Check out site http://www.arches.uga.edu/~godlas/sufism1.html if you are interested in a less well known sect of Islam - Sufism - a sect that has saints and was viewed by many theologians in the past as being close to Gnostic Christianity.
Herein lies much of the problem--"according to Muslim belief" they worship the same God. However, the same thing is often said by those who hold to New Age beliefs. In fact, many New Agers even insist they follow Christ. But when one examines who they believe Christ to be, it is immediately apparent that they do not in fact follow Christ. In the case of Muslims, they follow the writings of Mohammed, someone they insist is a prophet (or "the prophet") of God, yet this prophet is clearly not a true prophet in the Christian or Jewish faith. If one follows the teachings of a "false prophet" then does it not also follow that they in fact follow a "false god?"
I really should know more in order to deal with this question more knowledgeably, but I guess my main point is that we need to learn to be discerning about these issues. Particularly, we need to go beyond the claims of a particular religion and examine the truth of that religion according to the principles and objective truth of our Christian religion in order to determine whether or not the two religions are related.
Also, I think there is more than just the Qu'ran that forms the basis of Islamic beliefs--it's a place to start, but there are other books they follow in addition to that that need to be investigated.
-penny