Quick made his Major League debut on August 4, 1974, during a doubleheader between the Cincinnati Reds and San Diego Padres at San Diego Stadium.
As the home plate umpire in Game Six of the 1985 World Series, Quick called Jim Sundberg safe on Dane Iorg's ninth-inning single for the winning run in the Kansas City Royals' 2–1 victory over the St. Louis Cardinals.
The inning is best known for Don Denkinger incorrectly calling Jorge Orta safe on a ground ball leading off the inning; with the Royals trailing 3 games to 2 in the Series and 1–0 on the scoreboard, the call shifted the momentum both of the game and the Series for the Royals, who won 11–0 the next evening.
Quick was the home plate umpire for four no-hitters during his career: Tom Browning's perfect game on September 16, 1988, Jerry Reuss's no-hitter on June 27, 1980, Charlie Lea's on May 10, 1981 and Tommy Greene's on May 23, 1991.
Quick was also the home plate umpire for Nolan Ryan's 3000th career strikeout on July 4, 1980 and Steve Carlton's 300th career victory on September 23, 1983, and was the first-base umpire for Pete Rose's 4191st career hit on September 8, 1985, which put Rose in a first-place tie with Ty Cobb for the Major League lead.