It would be pretty awesome if it did. I know it’s a stupid question, but please indulge me.
ALSO- does the school offer badminton as a club sport? I know it has badminton courts but is badminton a club sport available there?
A simple internet search reveals that Boston University has a quidditch team but Boston College does not.
A quick search under the “club sports” tab of the BC Rec page reveals they do not have club badminton. It appears they do offer have an intramural tournament though.
There is no reason to ask questions here that can be searched on the internet.
No 
No quidditch at BC, although there is a Harry Potter club. You could absolutely start a quidditch club at BC though if you wanted to. There are a ton of people who like Harry Potter at BC and I can pretty much promise you there’d be people for you to play with.
That would cut down the traffic on this forum a lot!