I THINK they have the same God. If they do though, I do know that they view them differently. To my understanding, the Jewish version of God is more like a Father where as a Muslim's view of God is as a King. This is just my best guess because I am not a Muslim or a Jew.
Technically the God in Christianity is the same God (Allah) as in Islam (Muslim religion), which are both the same God in Judaism. The difference comes in certain core beliefs:
Christians believe in a triune God (one God made of three aspects - father, son and holy spirit) while Jews and Muslims do not.
Christians believe that Jesus was the son of God and the saviour as well as a prophet, Jews and Muslims do not.
Jews believe that the saviour has not yet come - this is not true for Christians and I don't think it is true for Muslims.
Muslims believe Muhammad is the greatest prophet, while Christians and Jews do not. And so on.
But yes, Jews, Christians and Muslims all believe in the same God, just different variations of this God.
As much as Christianity and Islam stem from Judaism, the direct contradictions of the texts paint pretty different characters. So, in name they do... Kind of... But the dramatic differences in their beliefs, and the different mandates they paint from their gods, seem to say differently.
One could say that, given god is supposed to be unerring and perfect, that in a bare sense Islam seems more consistent with Judaism than Christianity. The transition of an all knowing, perfect god from an eye for an eye to turn the other cheek seems polar opposite and impossible of a leap to make.
Does god change...?