Did St Peter have a rock in which he built the church of Jesus?

No.

This is the verse you are referring to:

Matthew 16:18 And I also say to you that you are Peter, and on this rock I will build My church, and the gates of Hades shall not prevail against it.

The big question is, "Who or what is the rock?" There is confusion from the fact that the Greek words for Peter and for rock are similar, but the meanings are different. The first, petros, means a stone or loose rock; the second, petra, means rock, such as a rocky ledge. So what Jesus really said was " ... you are Peter (stone), and on this rock I will build My church He did not say He would build His church on a stone but on a rock.

If Peter is not the rock, then what is?

If we stick to the context, ( back in verse 16 Peter confessed that Jesus was "the Christ the son of the living God.") the answer is that the rock is Peter's confession that Christ is the Son of the living God, the truth on which the church is founded.