The forward sustained the injury in the first period of a 5-4 loss to the Dallas Stars on Thursday when he scored his first goal and got his first assist in his second NHL game. He played through the injury the rest of the game.
"The injury he has, there's a window between week four and six that you can really get a gauge where he's at," Sabres coach Don Granato said. "It looks like it's going to be at least four."
The No. 8 pick in the 2020 NHL Draft, Quinn scored 35 points (18 goals, 17 assists) in 24 games with Rochester of the American Hockey League before being recalled Thursday.
"It is (unfortunate), especially you see him play the way he's played," Granato said. "I mentioned going to Rochester last Friday and watching him, he scored a goal very similar to the one he scored here, and he's been a scoring machine."
Quinn made his NHL debut Jan. 11 against the Tampa Bay Lightning before being reassigned to Rochester. The 20-year-old scored five goals in three AHL games after his return, including four in one game on Jan. 15.
"I don't worry about this (injury) stunting his development," Granato said. "I said when he left training camp that the kid, he's like a homing torpedo. I think was what I mentioned with him, and you see why I said that. He just keeps getting better. Every situation makes him better, he finds a way to continue to learn.
"I've always said as coaches, not me, coaches have always said the fastest learners are the ones that are going to become the best players. You have to have a degree of talent, which he has. He wants to learn and he learns fast. It's a bummer for us not to have him, but he's going to come out of this better, I just know he is."
Sabres forward Vinnie Hinostroza is also be out at least four weeks with a lower-body injury. Defenseman Will Butcher is day to day after sustaining an upper-body injury in the first period against the Stars.