ANSWER: Jesus does appear in the Quran, but He does not seem no where near as important to the average Muslim compared to Muhammad. That is, Muslims look upon him (Issa) nearly the way Christians perceive the prophets Isaiah or Moses. They believe he is an important true prophet, but not the Son of God. They certainly do not believe that Jesus is part of the Godhead and the Savior of all mankind.
Jesus, in the New Testament, is the central Personage of the book. The four Gospels are records of His actions and teachings in detail. By contrast, in the Muslim Quran, He is mentioned in only 74 verses out of a total of 6,236 verses, and 42 of those are indirect references (Ron Rhodes, "Reasoning from the Scriptures with Muslims," page 132). One verse in the Quran that mentions him denies He is both God and God's Son.
"The Messiah, Jesus, son of Mary, is only an apostle of God, and His word which He conveyed into Mary . . . Believe therefore in God and his apostles, and say not, 'Three:' (there is a Trinity) - Forbear - it will be better for you. God is only one God! Far be it from His glory that He should have a son!" (Sura 4:169 in Quran)
Because a Muslim, based on the Quran, believes God is only one Person, they adamantly deny the traditional Christian teaching that Christ is God. Sura 3:51 teaches the Lord was a man just like Adam when it states, "Verily, Jesus is as Adam in the sight of God. He created him of dust: He then said to him, 'Be' - and he was."
In contrast to the Quran, Scripture emphatically teaches that Jesus is not only the Messiah and the Son of God, He IS God (John 1:1, 14). He was both man and God at the same time while He walked the earth. A Muslim may also deny that the Lord died or that He was crucified, based upon Sura 4:155 - 156.
Perhaps Muhammad, whose principles are found in the Quran, was influenced by some kind of Gnostic teaching first taught centuries earlier. The four Gospels are very clear that Christ was killed by crucifixion, and that His death was not miraculously prevented. Rather, after He died, He was miraculously resurrected back to life again.
Ironically, it seems that the Muslim Quran contradicts itself in this regard, since Sura 4:48 states, "Remember when God said, 'O Jesus! Verily I will cause thee to die, and will take thee up to Myself and deliver thee from those who believe not."
In conclusion, the Muslim Quran (Koran) definitely mentions Jesus, but its teachings about him often contradict what the Bible teaches. Since the New Testament was written by first century followers of the Messiah, it is a very reliable witness to what he did, taught and what His followers recorded about him. It could be argued to be even more reliable a witness than the testimony regarding Muhammad, which was recorded or recited after his death by those who never knew Him personally.