On the one hand, the Quran insists that Muslims worship the God of Abraham. Surah 2:136 says:
Say, [O believers], "We have believed in Allah and what has been revealed to us and what has been revealed to Abraham and Ishmael and Isaac and Jacob and the Descendants [al-Asbat] and what was given to Moses and Jesus and what was given to the prophets from their Lord. We make no distinction between any of them, and we are Muslims [in submission] to Him." (Saheeh International)
Muhammad instructed his followers to say this to Jews and Christians: “We believe in that which has been revealed to us and revealed to you. And our God and your God is one; and we are Muslims [in submission] to Him” (Surah 29:46).
Nevertheless, Muslims also believe that the Bible has been corrupted, and false ideas about God have been inserted into the Scriptures. At the top of the list is the doctrine of the Trinity. The Quran is emphatic that the Christian concept of the Trinity is a false understanding of God.
They have certainly disbelieved who say, “Allah is the third of three.” And there is no god except one God. And if they do not desist from what they are saying, there will surely afflict the disbelievers among them a painful punishment. So will they not repent to Allah and seek His forgiveness? And Allah is Forgiving and Merciful. The Messiah, son of Mary, was not but a messenger; [other] messengers have passed on before him. And his mother was a supporter of truth. They both used to eat food. Look how We make clear to them the signs; then look how they are deluded. (5:73-75).From the Muslim point of view, the deity of Jesus and the doctrine of the Trinity constitute shirk, the Arabic term for a heretical departure from belief in the oneness of God by associating partners with Him.
Does the fact that Muslims reject the Trinity mean that the God of Islam is completely different from Christianity? Some writers think so. For example, one popular evangelical resource on Islam poses these questions: “Is Allah triune? If not, then we are not talking about the same God. Does Allah have a Son? If not...then we are not discussing the same God” (Unveiling Islam: An Insider’s Look at Muslim Life and Beliefs, p. 108).
At first glance these questions may seem to clinch the matter – until they are applied to Judaism. Jews do not believe in the Trinity, either, nor do they believe that Jesus is the son of God. Does this mean that Jews worship a different God than Christians? The apostle Paul didn't seem to think so (Acts 13:13-47; 22:1-14; 26:6). By the same standard, it is not consistent to argue that Muslims worship a different God simply because they are not Trinitarian and do not acknowledge Jesus is God’s Son. Other factors would have to be taken into account.
One such factor might be the radical otherness of the Islamic conception of God. Islam places such high emphasis on God's transcendence that it recoils at the mere though that God could be depicted as a father. “[He is] Originator of the heavens and the earth. How could He have a son when He does not have a companion [i.e., wife] and He created all things? And He is, of all things, Knowing.” (Surah 6:101; also see 72:3).
Yet even on this matter, we must acknowledge that the Jewish concept of God is in some respects similar to the Islamic view. To be sure, the Old Testament refers to God as "Father." But it does not do so frequently, and no one in the Old Testament ever addressed God as "Father" while praying to Him. As one New Testament scholar comments,
In the Old Testament, we are hard pressed to find much evidence of God being called “Father,” much less prayed to as Father, but this is the dominant way God is named and addressed in the New Testament, which is a reflex of the unique relationship that Jesus believed he had with God and could pass on to his followers to a lesser degree, such that they too could address God as “Abba, Father.” (Ben Witherington, The Indelible Image: The Theological and Ethical Thought World of the New Testament, Volume 2, p. 45)
So to be fair, if we are going to argue that Muslims worship a different God than Christians because they do not pray to God as “Father,” then we must apply the same reasoning to Jews. Or, if we believe that Jews and Christians worship the same God in spite of these serious disagreements about the Trinity and God’s Fatherhood, then to be consistent we must acknowledge that these issues alone do not determine whether Christians and Muslims worship a different God.
So here is my conclusion. Christians and Muslims share a common belief in the one God of Abraham, but each faith views the one God through a different set of lenses. And just like with a pair of glasses, the quality of those lenses will determine the clarity of vision - in some cases making a radical difference (as when I try to use my wife's glasses). As you can see from the citations from the Quran I provided, the Islamic view of God is shaped by a reactionary response to polytheism, creating a distorted characterization of Christian teaching, and a blurry view of God. Whereas Muslims believe the most direct revelation of God is found in a book (the Quran), Christians believe the most direct revelation of God is found in a person, Jesus. And it is through Jesus that I believe we see God most plainly.