Funny that that is not really known, is it? Diaries mention a Quarrymen performance in the Cavern in August, but without Paul. Pauls debut comes not earlier than October. This is what Murashev says:
1957 Oct 18
Friday Evening: Performance of the Quarrymen in a ball held by the Conservative Club, at the New Clubmoor Hall, Broadway, Liverpool. Paul's debut. Paul plays for John a couple of tunes he had composed, one of them being `I Lost My Little Girl'.