The wikipage for the pilgrimage gives a lot of detail, but it never says a specific age in regards to accurately pinpointing when. I think it says 'young adulthood' as being the right time for a quarian to go on their pilgrimage. Mine is specifically going on his at 19, but I assume they deem young adulthood around 18-20, though mine is specifically on his for around 3 years, I think I specified.

EDIT: I'll actually leave that until I get to actually posting my guy.