Quinn was dominating the OHL during the 1983-84 season, with 59 points in 24 games when he was summoned to join the Calgary Flames. In his rookie season with the club, Quinn scored almost a point-a-game, recording 52 points in 54 games. He maintained that strong play in the playoffs and was a vital cog in Calgary's run to the 1986 Stanley Cup Final.

On November 12, 1986, Quinn was traded to the Pittsburgh Penguins. It was in Pittsburgh that Quinn would have the most success of his NHL career, recording seasons of 28, 40 and 34 goals respectively.

January 8, 1990, Quinn was on the move again. This time traded to the Vancouver Canucks along with Dave Capuano and Andrew McBain in exchange for Rod Buskas, Barry Pederson and Tony Tanti. Quinn would spend a season and half with the Canucks before being shipped to the St.Louis Blues at the NHL trade deadline. At season's end the Blues traded him to the Philadelphia Flyers.

Quinn would spend just on season with the Flyers, appearing in 67 games while recording 11 goals and 26 assists. He would sign as a free agent with the Minnesota North Stars prior to the 1992-93 season. Quinn was released by the North Stars after just 11 games and he finished the season with SC Bern of Switzerland's top league.

After agree to terms in the summer, Quinn posted 31 points in 44 games for the Los Angeles Kings during the 1994-95 season, but he was on the move again the following summer, signing as a free agent with the Ottawa Senators. In 1995-96, he tallied 24 points in 28 games for the Senators before being traded mid-season to the Philadelphia Flyers. In 35 games the Flyers Quinn scored seven times and added 14 assists.

On July 17, 1996, signed as a free agent with the Pittsburgh Penguins, but would retire after just 16 games.

After his playing career, Quinn turned to his other sporting pleasure, golf. He won the American Century Celebrity Golf Championship Tournament in Lake Tahoe five times and caddied in all four majors including The Masters for Ernie Els in 2010.