Markus Bockmuehl presents that in Luke 22.31-32, we are presented with a problem of when was Peter's turning. Luke in Luke and Acts does not answer this question.
So it seems that Peter at least in the Biblical Text that we have no account of Peters turning or conversion.
Bockmuehl looks to the understanding of this problem in Petrine memory in the first three centuries of the Church
In Catacombs visual representation of Christ and Peter with a cock between them, and on a pillar. Cock is then clearly significant raised on a pedestal.
Also on sarcophagi, on the ground but more than life size.
What happens to the cock, what happens to the cock and why is it there?
Prediction of Peters denial, is a surface interpretation of this image. If this is so no conversion or call here. but this doesn't explain the frequency nor visual significance given to the cock.
Early Christians interprets the cock in terms of the redemption new life, cock as announcing the dawn of new life and renewal and repentance.
The image telescopes the fall and peters return and renewal.
Peter's conversion continued here