The name of Jesus is scattered throughout the Qur'an. He is mentioned by name 25 times. Therefore all Muslims believe he existed as do Christians. Oh and Mary is the only woman's name mentioned in the Qur'an and the 19th Qur'anic chapter is called Mary.
The message of Islam was given and followed by ALL the PROPHETS.
And We sent not before you any messenger except that We revealed to him (MUHAMMAD) that, "There is no deity except Me, so worship Me."
Qur’an 21:25 See also Qur’an 21:107, Exodus 20:3, Deuteronomy 5:7
84. We gave him (ABRAHAM) ISAAC and JACOB: all (three) guided: and before him, We guided NOAH, and among his progeny, DAVID, SOLOMON, JOB, JOSEPH, MOSES, and AARON: thus do We reward those who do good:
85. And ZAKARIYA and JOHN, and JESUS and ELIAS: all in the ranks of the righteous:
86. And ISMA'IL and ELISHA, and JONAS, and LOT: and to all We gave favour above the nations:
87. (To them) and to their fathers, and progeny and brethren: We chose them, and we guided them to a straight way.
88. This is the guidance of Allaah: He giveth that guidance to whom He pleaseth, of His worshippers. If they were to join other gods with Him, all that they did would be vain for them.
Qur’an 6:84-88
And those who disbelieved say, "Why has a sign not been sent down to him from his Lord?" Say, [O Muhammad], "Indeed, Allaah leaves astray whom He wills and guides to Himself whoever turns back [to Him] -
Qur’an 13:27