As a child, Quinn played for the Southwest region, representing the state of Louisiana at the 2008 Little League World Series, where he threw a no-hitter in the opening round. The team finished in fourth place, losing the U.S. Championship to Hawaii and the consolation game to Japan. Quinn attended Barbe High School in Lake Charles, Louisiana. Along with football, Quinn also played baseball and ran track. During his senior year at Barbe, he recorded 1,967 receiving yards and 23 touchdowns. He finished his high school career with 70 touchdowns and 6,566 receiving yards, making him the national all-time career leader in receiving yards. Quinn committed to play football for the LSU Tigers in August 2013, choosing the Tigers over Auburn, Clemson, and Texas.
Quinn began his college football career at LSU in 2014. He played in 13 games for the Tigers as a true freshman, finishing the season with 17 receptions for 193 yards. As a sophomore in 2015, he played in all 12 of LSU's games. He caught five passes for 83 yards. He left the team after the season.
Quinn transferred to Southern Methodist University prior to the 2016 season. Due to NCAA transfer rules, Quinn sat out the 2016 season and redshirted. As a redshirt junior in 2017, Quinn played in all 13 games for SMU, catching 114 passes for 1,236 yards and 13 touchdowns. After the season, Quinn declared for the 2018 NFL Draft.
Professional career
2018
Quinn was drafted by the Washington Redskins in the 7th round of the 2018 NFL Draft, making him Mr. Irrelevant. He suffered a high-ankle injury during the first game of the season and was subsequently placed on injured reserve a few days later. He was activated off injured reserve on November 14, 2018. He was placed back on injured reserve on December 5, 2018 after re-injuring his ankle. He finished his rookie year with 75 receiving yards and 1 touchdown on 9 receptions.
2019
In Week 1 against the Philadelphia Eagles, Quinn caught 4 passes for 33 yards and a touchdown as the Redskins lost 27-32. In the Week 13 win against the Carolina Panthers, he was removed from the game sustaining a helmet to helmet hit on a punt return and was diagnosed with a concussion. After missing the next two games due to concussion symptoms, the Redskins placed Quinn on injured reserve on December 17, 2019.