"No one who denies the Son has the Father; whoever acknowledges the Son has the Father also." (1 John 2:23 NIV)
Long before Muhammad was born, Arabic Christians were already referring to God as Allah — and millions continue to do so today. The Allah of Islam, however, is definitely not the God of the Bible; for while Muslims passionately defend the unity of God, they patently deny His triunity. They recoil at the notion of God as Father, reject the unique deity of Jesus Christ the Son, and renounce the divine identity of the Holy Spirit.
First, while Jesus taught His disciples to pray "Our Father in heaven," devotees of Muhammad find the very notion offensive. To their way of thinking, calling God "Father" and Jesus Christ "Son" suggests sexual procreation. According to the Qur'an, "It is not befitting to (the majesty of) Allah that He should beget a son" (Sura 19:35); and Allah "begetteth not, nor is he begotten" (Sura 112:3). The Bible, however, does not use the term "begotten" with respect to the Father and the Son in the sense of sexual reproduction but rather in the sense of special relationship; thus, when the apostle John speaks of Jesus as "the only begotten of the Father" (John 1:14 KJV, emphasis added), he is underscoring the unique deity of Christ. John goes on to state, "No one has ever seen God, but God the One and Only, who is at the Father's side, has made him known" (John 1:18 NIV). When the apostle Paul likewise refers to Jesus as "the firstborn over all creation" (Colossians 1:15 NIV, emphasis added), he is emphasizing Christ's preeminence or prime position as the Creator of all things (cf. vv. 16–19). Christians are sons of God through adoption; Jesus is God the Son from all eternity.
Muslims, furthermore, dogmatically denounce the Christian declaration of Christ's unique deity as the unforgivable sin of shirk. As the Qur'an puts it, "Allah forgiveth not (the sin of) joining other gods with Him; but He forgiveth whom He pleaseth other sins than this" (Sura 4:116). Muslims readily affirm the sinlessness of Christ, however, they adamantly deny His sacrifice upon the cross and subsequent resurrection. In doing so, they deny the singular historical fact that demonstrates that Jesus does not stand in a long line of peers from Abraham to Muhammad, but is God in human flesh. The Qur'anic phrase, "Allah raised him up" (Sura 4:158) is taken to mean that Jesus was supernaturally raptured rather than resurrected from the dead. In Islamic lore, God made someone look like Jesus, and this look-alike was crucified in His place. In recent years, the myth that Judas was crucified in place of Jesus has been popularized in Muslim circles due to the propagation of a late-medieval work titled The Gospel of Barnabas. Against the weight of historical evidence, the Qur'an exclaims, "they killed him not, nor crucified him, but so it was made to appear to them" (Sura 4:157).
Finally, in addition to rejecting the divinity of Jesus, Islam also renounces the divine identity of the Holy Spirit. Far from being the third person of the triune God who inspired the text of the Bible, Islam teaches that the Holy Spirit is the archangel Gabriel who dictated the Qur'an to Muhammad over a period of 23 years. This is ironic considering that Islam also identifies the Holy Spirit promised by Jesus in John 14 as Muhammad. The Bible, however, roundly rejects such corruptions and misrepresentations. The Holy Spirit is neither an angel nor a mere mortal; rather, He is the very God Who redeems us from our sins and will one day resurrect us to life eternal (e.g., Acts 5:3–4; Rom. 8:11).
For further study, see Timothy George, Is the Father of Jesus the God of Muhammad? (Grand Rapids: Zondervan, 2002).