The Scripture does not give us this detail. What is interesting, though, is that Epaphras, a fellow prisoner of Paul, and Onesimus both came from Colossae. Epaphras planted the church in Colossae, and the church met in the house of Philemon (where Onesimus was a slave). It is pretty certain that Epaphras would have known Onesimus from that time. And therefore it could well be that Epaphras recognized Onesimus somewhere and that this was Onesimus's line of contact with Paul. Whatever it may be, by God’s grace Onesimus heard the good news about Jesus.
10 I appeal to you for my child, Onesimus, whose father I became in my imprisonment.