Quine was originally selected by the Detroit Red Wings in the third round (No. 85) of the 2011 NHL Draft. He wasn't signed and re-entered the Draft in 2013, when he was selected in the sixth round (No. 166) by the New York Islanders.
The versatile forward played for three teams (Kingston, Peterborough, Belleville) in the Ontario Hockey League from 2009-13 and had 218 points (90 goals, 128 assists) in 252 games. He signed an amateur tryout contract with Grand Rapids of the American Hockey League on March 26, 2012, and played three games, finishing with one assist.