The team said the 31-year-old was injured at his home in Miami, suffering a laceration on his left thumb that required five stitches.
"This morning in Chicago, Quintana underwent microscopic surgery on his left thumb to further determine the extent of his injury," the team said Thursday in a statement. "The procedure identified a lacerated digital sensory nerve in his left thumb, which was surgically repaired."
Quintana is expected to start throwing again in two weeks, and then the team will determine how long he will be out.
"There's a best-case scenario in which it heals quickly and his thumb feels good and he can resume a pretty rapid ramp up from that point," president of baseball operations Theo Epstein said. "He's been built up, is pretty far along. But there's certainly another scenario in which the nerve takes longer to heal and is going to be significantly delayed. We really can't forecast it. We just have to wait and see and hope for the best."
Alec Mills could take Quintana's spot in the rotation. The 28-year-old right-hander had a 2.75 ERA in four starts and nine appearances last season. Right-handers Adbert Alzolay and Colin Rea are also in the mix.
"This injury certainly challenges our depth in an area where we had some concerns already about our depth," Epstein said. "It's going to be important for certain players to step up and make the most of this opportunity."
He added: "Someone like Alec Mills is well positioned to capitalize on this opportunity and establish himself as a starting pitcher in this league."
Quintana was 13-9 with a 4.68 ERA last season, his third with the Cubs. After he had an All-Star season in 2016, the Cubs acquired him from the White Sox in a July 2017 deal that saw young slugger Eloy Jimenez head to the South Side.
Chicago finished third in the NL Central last season at 84-78 and missed the playoffs for the first time since 2014.
"Given the situation that we're in as an industry starting this season with so much uncertainty surrounding us, it would be foolish for us for everything to go smoothly and to have all our players available to us at all times," Epstein said. "We didn't necessarily see this circumstance coming with Q. It's something you can't anticipate. We know there are going to be absences that we have to fill, and there's going to be adversity we're going to have to overcome."
ESPN's Jesse Rogers and The Associated Press contributed to this report.