The team announced Tuesday that Quinn sustained the injury during practice, and he is expected to undergo surgery in Dallas.
"The relief is he's going to be back and he's going to be fine," Jones told reporters. "He'll be able to get his work in and do what he needs to play the Giants. I don't think we're looking at him missing any [regular-season] time here."
Quinn sustained the injury while competing against Pro Bowl left tackle Tyron Smith during a one-on-one pass rush drill, according to ESPN. It's unclear whether the defender got his hand caught in Smith's elbow brace or face mask.
Cowboys trainers inspected Quinn before leading him to the locker room for further examination.
The then-St. Louis Rams selected Quinn in the first round of the 2011 NFL Draft out of North Carolina. The two-time Pro Bowl choice spent last season with the Miami Dolphins before being traded to the Cowboys in March.
Quinn has 69 sacks in his career, including 6.5 sacks with the Dolphins last season.