The Rangers fired former head coach Alain Vigneault last month after going a disappointing 34-39-9 and missing the playoffs in 2017-18.
Quinn has been the head coach at Boston U. since 2013-14.
In his five seasons with the Terriers, Quinn went 105-67-21 with four trips to the NCAA tournament.
That included a run to the Frozen Four in 2014-15 that saw the Jack Eichel-led Terriers lose to Providence in the National Championship Game.
Before becoming the head coach at Boston University, Quinn was an assistant at Northeastern, Nebraska-Omaha and Boston University.
He then served as head coach of the AHL's Lake Erie Monsters and was an assistant for the Colorado Avalanche before returning to Boston U. to become head coach.
Vigneault was the Rangers head coach for five seasons, and he led the Blueshirts to the playoffs in each of his first four campaigns at the helm.
In 2013-14 he brought them to the Stanley Cup Final, where they lost to the Los Angeles Kings.
Overall, New York went 226-147-37 under Vigneault.
Although the Rangers took a significant step back last season, Quinn will inherit a roster with no shortage of talent.
If veteran goalie Henrik Lundqvist can bounce back from a down year to supplement a forward group that includes Mats Zuccarello, Mika Zibanejad, Chris Kreider, Jimmy Vesey and Ryan Spooner, the pieces are in place for a quick turnaround in the Big Apple.