Yes, Jesus did say he was the Messiah. He said he was Messiah to the Samaritan woman at the well at Sychar. John 4:25-26:
The woman said to him, “I know that Messiah is coming (he who is called Christ). When he comes, he will tell us all things.” (26) Jesus said to her, “I who speak to you am he.”
Note that “Messiah” is the Hebrew equivalent of the Greek word “Christ”.
Jesus acknowledged being the Christ, after Peter’s confession in Matthew 16:15-17:
(15) He [Jesus] said to them, “But who do you say that I am?” (16) Simon Peter replied, “You are the Christ, the Son of the living God.” (17) And Jesus answered him, “Blessed are you, Simon Bar-Jonah! For flesh and blood has not revealed this to you, but my Father who is in heaven.
Soon after Jesus’ baptism and commencement of his public ministry, he went to the synagogue at his home town of Nazareth and stood up to read. He was given the scroll of the prophet Isaiah. Jesus unrolled the scroll and found the place and read:
Luke 4:18-21 “The Spirit of the Lord is upon me, because he has anointed me to proclaim good news to the poor. He has sent me to proclaim liberty to the captives and recovering of sight to the blind, to set at liberty those who are oppressed, (19) to proclaim the year of the Lord’s favor.” (20) And he rolled up the scroll and gave it back to the attendant and sat down. And the eyes of all in the synagogue were fixed on him. (21) And he began to say to them, “Today this Scripture has been fulfilled in your hearing.”
Jesus was reading from the Septugint, Isaiah 61:1 and part of verse 2. This was a proclamation of who he was – the Messiah. And how did the people of Nazareth react to the gracious words he spoke? Luke 4:28-29:
When they heard these things, all in the synagogue were filled with wrath. (29) And they rose up and drove him out of the town and brought him to the brow of the hill on which their town was built, so that they could throw him down the cliff.
He claimed to be the son of God in John 10:36:
do you say of him whom the Father consecrated and sent into the world, ‘You are blaspheming,’ because I said, ‘I am the Son of God’?
To claim to be the son of God was tantamount to claiming to be the Messiah. See Psalm 2:6-7:
“As for me, I have set my King on Zion, my holy hill.” (7) I will tell of the decree: The LORD said to me, “You are my Son; today I have begotten you.
At Jesus’ trial, he made several statements, which in effect were a claim to be the Messiah. He didn’t come straight out and say this, but on several occasions, when he was asked if he was the Christ or the King of the Jews, he replied “You have said so”. This expression means – “as you say, I am …”. We know this because in Matthew 26:25 when Judas asks Jesus if it is he who would betray him, Jesus replies, “You have said so.”
Some of the passages about the trial of Jesus, where he acknowledges Messiahship or Kingship:
But Jesus remained silent. And the high priest said to him, “I adjure you by the living God, tell us if you are the Christ, the Son of God.” (64) Jesus said to him, “You have said so. But I tell you, from now on you will see the Son of Man seated at the right hand of Power and coming on the clouds of heaven.” (65) Then the high priest tore his robes and said, “He has uttered blasphemy. What further witnesses do we need? You have now heard his blasphemy. (66) What is your judgment?” They answered, “He deserves death.” (67) Then they spit in his face and struck him. And some slapped him, (68) saying, “Prophesy to us, you Christ! Who is it that struck you?”
Now Jesus stood before the governor, and the governor asked him, “Are you the King of the Jews?” Jesus said, “You have said so.”
And Pilate asked him, “Are you the King of the Jews?” And he answered him, “You have said so.”
Jesus answered, “My kingdom is not of this world. If my kingdom were of this world, my servants would have been fighting, that I might not be delivered over to the Jews. But my kingdom is not from the world.” (37) Then Pilate said to him, “So you are a king?” Jesus answered, “You say that I am a king. For this purpose I was born and for this purpose I have come into the world–to bear witness to the truth. Everyone who is of the truth listens to my voice.”