According to the text Jesus called Himself, "Son of Man" in verse 13. Without verbal prompting Peter elevated the Son of Man to Son of God by verse 16.
Let's look...
13 When Jesus came to the region of Caesarea Philippi, he asked his disciples, “Who do people say the Son of Man is?”
14 They replied, “Some say John the Baptist; others say Elijah; and still others, Jeremiah or one of the prophets.”
15 “But what about you?” he asked. “Who do you say I am?”
16 Simon Peter answered, “You are the Messiah, the Son of the living God.”
17 Jesus replied, “Blessed are you, Simon son of Jonah, for this was not revealed to you by flesh and blood, but by my Father in heaven.Notice that Jesus NEVER says he is the Son of Man, nor does Peter say that the Son of Man is the Son of God. This is a classic example of Christians stringing things together and claiming the Bible says something it doesn't. Jesus asks about the Son of Man and then about himself. Two diferent questions, two different answers with Jesus never claiming to be either nor claiming to be God.