The 31-year-old Colombia native struggled throughout the 2019 campaign. He posted career-worst numbers in both ERA (4.68) and WHIP (1.39) while striking out 152 batters in 171 innings across 32 appearances.
Quintana said in spring training he was hopeful he could produce a bounce-back year in 2020.
"I want to be happy with what I'm doing on the mound," he told reporters. "I want to get my career in a good spot and show all the fans at Wrigley how I can be. I want everyone to feel excited when I'm on the mound."
The 2016 All-Star selection joined the Cubs in a 2017 trade with the crosstown rival Chicago White Sox. He provided a boost to the team's rotation down the stretch that year with a 3.74 ERA in 14 starts, and he compiled a 4.03 ERA in 32 outings during the 2018 season.
Quintana was set to serve as Chicago's No. 4 starter behind Yu Darvish, Jon Lester and Kyle Hendricks on one of the National League's deepest starting staffs.
He faces a tight timeline to make himself available for the start of the season, which is tentatively scheduled for July 23-24 amid the coronavirus pandemic.
Alec Mills and Jharel Cotton are among the options to fill the void should Quintana open the campaign on the injured list.