Scoring 52 goals in 62 games for Ottawa in the Ontario Hockey League during the 2019-20 season was enough to convince the Buffalo Sabres that Quinn was the player they wanted when their turn came around in the first round of the 2020 NHL Draft. They selected the 6-foot right wing with the No. 8 pick on Oct. 6, 2020 and signed him to an entry-level contract less than six weeks later.

Quinn was second in the OHL with 52 goals and became the fourth OHL player from 2010-20 to score at least 50 goals in his draft year, joining Jeff Skinner (50 goals for Kitchener in 2010), Alex DeBrincat (51, Erie, 2016) and Arthur Kaliyev (51, Hamilton, 2019).

He scored 32 points (12 goals, 20 assists) in 61 games in 2018-19 and credited extensive offseason workouts which helped him get stronger when he had his breakout season 2019-20.

Quinn made his NHL debut Jan. 11, 2022, a 6-1 loss to the Tampa Bay Lightning. He scored his first NHL goal in a 5-4 loss to the Dallas Stars on Jan. 20. He set an NHL career-high with 39 points (15 goals, 24 assists) in 74 games of the 2024-25 season and signed a two-year, $6.75 million contract (average annual value of $3.375 million) with Buffalo on June 27, 2025, in lieu of becoming a restricted free agent.