There are some great discussions going on at Stand To Reason and the Denver Seminary on this specific topic, but I think the best way to answer this question is not necessarily if it is The same God, but rather what are the paths taken to get to such a God...
...I must preface by saying that the identity of God is of utmost importance because Christians believe in the Trinity yet Muslims do not, so we need a way to help deduce the identity of The Christian God and the Islamic God, and this deduction comes through the path...the path in which we get to God...
For instance, John 14.6 states, "I am the way, the truth, and the life!" Jesus answered. "Without me, no one can go to the Father." So let's say, for arguments sake, that Christians and Muslims worship the same God, then we must focus on how to get to Him...
Imagine you and your friend want go to the moon one day. You decide to take NASA's path and head up through the earth's atmosphere, but your friend thinks he can get there by heading East on Interstate 10...You both want to go the same place, but only one of you has the correct path...
...therefore, since Christians believe that Jesus is The One and Only Son of God, and Jesus claims that He is the only way to God, then Christians and Muslims definitely worship a different God...how do I know this? Because if you believe that there is more than one path to the same God, then you now believe in religious pluralism, and this goes against one of the fundamental laws of logic, the law of non-contradiction: something can not be true and not true at the same time...for example, Jesus can't be The only way to God, and not the only way to God at the same time and in the same context.
Bottom line: you can accept the Christian view of The Trinity and that Jesus is The Only Way to God, or you can reject it, but you must understand one MAIN issue that separates all beliefs... Jesus Christ changes everything...Lee Strobel puts it this way:
"In short, if the Resurrection is false, then Christianity is refuted. But if it's true, then regardless of what any world religion teaches, Jesus is the one-and only Son of God. And that changes everything." (see 1 Corinthians 15)
