The Friday prayer, known as Salat al-Jumu'ah, began after the Prophet Muhammad (peace be upon him) and his followers migrated from Mecca to Medina.
Specifically, it's widely believed to have started when the Prophet Muhammad (peace be upon him) was on his way to Medina, before he actually arrived. He stopped in a place called Quba, and then continued on to a place called the Valley of Ranuna, where he led the first Friday prayer.
So, the generally accepted answer is: Shortly after the Hijra (migration to Medina), around 622 CE.