No. The god of Islam is NOT the God of Christianity. P E R I O D.
Lets say I'm an author, call me Dan. I decide to write a book about Jesus. In my book, I indicate that Jesus didn't really die but he married His beloved disciple, John, who really wasn't John at all, but Mary Magdalene and this Jesus fathered to children who grew up in France and fathered all of Europes royal families. Is the Jesus spoken here the same Jesus? Is it just an "imperfect understanding" of Jesus? No. It is a made-up Jesus who is given attributes of the real Jesus - attributes stolen from the Bible.
Would it be right for me to say, we worship the same Jesus as the Davinci Code readers. If you answer affirmative to that question then you have missed the point of Christianity entirely.
If one looks at the Bible as history, then it is the history of a progressive revelation. Beginning with Abraham, it is the revelation of God to a particular people. We Christians believe that that divine history found its fulfillment in Jesus Christ. The Jews deny this, so you would say that the God of Jesus is not the God of the Jews? To the extent that the Koran does teach us about Abraham and teach us what the Bible does, then it is true.