Quillan began his collegiate career for the Quinnipiac Bobcats during the 2021–22 season, where he recorded two goals and seven assists in 37 games.

During the Frozen Four at the 2023 NCAA Division I men's ice hockey tournament, Quillan scored two goals in a victory over Michigan and help Quinnipiac advance to the national championship game.

[3][4] During the championship game, he scored the game-winning goal ten seconds into overtime to defeat Minnesota and win the program's first NCAA title.

[13][14] Starting his career off with the Toronto Marlies, Quillian played well in his debut season, and was rewarded with a call up to the Maple Leafs roster on January 22, 2025 following an injury to Max Pacioretty.

[15] His ice-time was limited to just over 5 minutes, as in the first period Quillan was struck by a knee-on-knee hit from Senators forward Nick Cousins, although he remained in the game.