No, it didn't come from Peter. That's a late tradition based on
-The 2nd century Acts of Peter in which Peter goes to Rome and performs all sort of absurd miracles like raising fish from the dead and battling Simon Magus in the sky and making dogs talk. The claim that the Catholic church began with Peter
-Pope Anecitus in the late 2nd century who during an early power struggle over when Jesus died claimed Peter founded the Roman church to back up the day Roman Catholics celebrated, Nisan 13, to try to trump the Asian churches who claimed John as their founder and said Jesus died on Nisan 14; king trumps queen, Peter trumps John, both being equally bullshit claims.
Go find me a source before Anicetus or Acts of Peter who claims Peter founded the church at Rome. You won't find one because he didnt. Everybody and their brother traced their authority to one apostle or disciple or another by the mid-2nd century.