"Didn't Q also send them across a vast gulf of space?"
In Star Trek TNG episode 42, original airdate week of 5/8/89, episode title "Q Who?", Q seeks to join the Enterprise crew. When Picard declines to allow it, Q, in a fit of pique, hurls the Enterprise instantaneously half-way across the galaxy. There, the first encounter takes place with the notorious Borg, and, although eventually Q bails out Picard and his Enterprise crew, the stage is set for subsequent Borg misadventures.
Yes, but the Borg did give us seven of nine.