Conley suffered the injury when he was inadvertently elbowed in the face by Portland Trail Blazers guard C.J. McCollum during Game 3 on Saturday.
After being hit, Conley remained on the floor as the match continued. He was eventually taken to the locker room to get an X-ray before travelling to a medical facility in Portland.
Conley flew back to Memphis on Sunday afternoon and underwent surgery on Monday morning.
"We don't know a lot, but the procedure was good and we'll see how it goes from here," Grizzlies coach Dave Joerger told reporters. "We've been trying to patch it together all year. We've had a lot of injuries, and one thing this team has done this season is it's stuck together."