Many have answered yes. In his book, Ecumenical Jihad, I think Peter Kreeft argues essentially that. In other words, both Christians and Muslims are theists and agree on a number of attributes God (or Allah) must have, being both infinite and in some sense personal, even if transcendent. Their understandings of God may not overlap entirely, but it's the same God, nonetheless, Whom they worship.
Other voices have been answering no. The Jewish radio talk show host, Michael Medved recently responded to the claim that Christian violence is comparable to Muslim terrorism by stating: "... no nations or prominent church groups promote or applaud violence in the name of Jesus, but several nations and many leading voices in Islam endorse violence in the name of Allah. Radical Islam stands alone among contemporary religious sects in suggesting that the slaughter of innocent women and children in suicide attacks will bring you closer to God."
This might suggest that the Christian understanding of God is different from that of radical Islam, and invites more explicitly a discipleship of love than Islam; but this still doesn't answer the question whether the God of Islam and Christianity is the same God.
Just today, however, Gerald R. McDermott published a piece, which the leftist pundits will likely tar and feather with accusations of hate speech: "No, the God of the Qur’an is Not the God of the Bible" (On the Square, June 4, 2013). Excerpts:
Yale theologian Miroslav Volf answers the question in a recent book (Allah: A Christian Response) with a nuanced but insistent Yes: Christians and Muslims do indeed worship the same God. In a review of Volf’s book, Baylor historian Thomas Kidd faults Volf for sidestepping the question of salvation—and therefore the question of true worship—and for not being critical enough in his evaluation of the identity of the God or gods of these two religions.[Hat tip to E. Echeverria]
Kidd is quite right; indeed, there are deeper problems with Volf’s thesis. His argument for the identity of the Muslim and Christian Gods collapses under its own weight. Volf’s own logic underscores what the Qur’an itself suggests—that the God of the Qur’an is radically different from the God Christians worship.
... Even Muslim scholars recognize that none of [the] five verses [in the Qur'an referencing "love" in connection with God] constitutes a command to love God. In his 1960 study, The God of Justice: A Study in the Ethical Doctrine of the Qur’an, Muslim scholar Daud Rahbar insisted that “the Qur’an never enjoins love for God.”
... Another difficulty is that there simply is no command to love one’s neighbor in the Qur’an. One can talk about love for neighbor in the Islamic tradition, but not as something commanded by the God of the Qur’an....
... All this is not to say that no Muslims can be saved by the person and work of Jesus Christ. Nor is it to suggest that Muslims never make contact with the true God. For Scripture attests that God graciously reaches out to those who have faulty notions of him. But we have to conclude nonetheless that the God of the Qur’an is a very different God from the God of the Bible.