Drafted by the Kingston Frontenacs in the first round (2nd overall) of the 2009 Ontario Hockey League (OHL) Priority Selection, Quine played 252 games in the OHL with the Kingston Frontenacs, Peterborough Petes, and Belleville Bulls, registering a total of 90 goals and 128 assists for 218 points, and 218 penalty minutes. Following the 2010–11 OHL season, Quine was selected by the Detroit Red Wings in the 3rd round of the 2011 NHL Entry Draft however, he did not sign with the Red Wings and continued to play major junior hockey in the OHL. Following the 2012–13 OHL season, Quine again heard his name called in the NHL Entry Draft, this time by the New York Islanders who picked him in the 6th round.