just to add another perspective
Death wish episode was on stardate 49301.2 which is April 20, 2372, Time: 05:44:26
Q states that Quinn has been locked up for 300 years
2372 - 300 = 2072 the year Quinn was locked up
first contact was April 5, 2063
Quinn must have interfered somehow when the Borg tried to assimilate Earth. And since Quinn appears to be a Q that is sincerely helping humans out (as evidenced by the other witnesses) then Quinn's character would suggest he would try to help humanity.
Q: A slight accident. Oh, let me guess. You were trying to commit suicide. Now you see why we've locked him up for the last three hundred years
Captain's log, stardate 49301.2. We have assigned quarters to our new passenger, who has entered his name on our crew manifest as Quinn. I am anxious to engage him in ship activities as soon as possible.
The First Contact took place on the evening of April 5, 2063, when a Vulcan survey ship, the T'Plana-Hath, having detected the warp signature of the Phoenix, touched down in Bozeman, central Montana