The Gators are scheduled to begin preseason practice on Friday.
"Jeff had acute appendicitis and fortunately the medical staff caught it very quickly," Florida coach Will Muschamp said in a statement. "His surgery went well and how long he is out will be determined by how his body responds, which could be two weeks."
With Driskel out, redshirt junior Tyler Murphy and redshirt freshman Skyler Mornhinweg will be the Gators' top two quarterbacks when preseason practice begins and likely will split first-team reps. Neither has thrown a pass in a game in their respective careers.
Florida also has four other quarterbacks on the roster -- walk-ons Chris Wilkes, Ryan McGriff and Jacob Guy and incoming freshman Max Staver. Wilkes joined the team in May after playing professional baseball for five years.
The 6-foot-4, 237-pound Driskel threw for 1,646 yards and 12 touchdowns with five interceptions and ran for 413 yards and four touchdowns as a sophomore last season after beating out Jacoby Brissett for the starting job.