The changes come a week after the Rangers named Chris Drury their new general manager and president after firing GM Jeff Gorton and president John Davidson.
Team owner James Dolan released a statement saying the team lacked "overall spirit and culture" under the former executives.
The club issued a letter to fans in 2018 outlining plans for a full-scale rebuild and signed Quinn to a five-year contract shortly afterward. Since then, the Rangers added MVP candidate Artemi Panarin via free agency as well as lottery picks Alexis Lafreniere and Kaapo Kakko.
The Rangers finished in fifth place in the East Division this season with a 27-23-6 record. They're in 15th place in the league's overall standings.