Yes, Jesus fulfills Old Testament Messianic prophecies.
For example, Isaiah 7:14 provides that a young maiden or virgin will gives birth to a son, whose name will be Emmanuel (“God is with us”), thereby anticipating the Incarnation of Jesus, born of the Virgin Mary (Matt. 1:18-23; Luke 1:26-38).
Similarly, there is Isaiah 9:6:
For to us a child is born,
to us a son is given;
and the government will be upon his shoulder,
and his name will be called
“Wonderful Counselor, Mighty God,
Everlasting Father, Prince of Peace.”
This passage also anticipates the Incarnation, with the fulfillment being seen in the aforementioned Infancy narratives of Matthew 2 and Luke 1, as well as in John 1:1-3 and John 3:16-17.
The Messiah is also prophesied to be born in Bethlehem (Mic. 5:2; see 2 Sam. 2:6), and that is where Jesus is born (Matt. 2:6). In addition, the Messiah is prophesied to be an ensign to all of the nations (Isa. 11:1-10) who will restore the kingdom of Israel (Am. 9:11-12). And Jesus receives the throne of his father David and his kingdom endures forever (Luke 1:32-33). And Jesus makes it his mission to make disciples of all nations (Matt. 28:18-20).
For more information, see our articles on Isaiah 7:14, Jesus as a second Moses, how Scripture affirms that Jesus is God, and that Jesus is not a false prophet.