Where did The Beatles perform the song Hey Jude and when did they perform it?

Probably the first performance by The Quarrymen (the earliest band formed by Lennon) was in July 1957 at a garden fete in Woolton, Liverpool. The Beatles became a "serious" band in 1960 when they obtained a booking to play in Hamburg at the Indra club; their first professional performances could be argued to be there.