Quinn is a third-generation member of a family involved in baseball management since the turn of the 20th century. His grandfather J.A. Robert Quinn at one time was an owner of the Boston Red Sox and Boston Braves and general manager of the St. Louis Browns and Brooklyn Dodgers. His father, John J. Quinn, was GM of the Braves in Boston and Milwaukee and the Philadelphia Phillies between 1945 and 1972. A brother-in-law, Roland Hemond, was also a longtime baseball executive and general manager of the Chicago White Sox and Baltimore Orioles. And Quinn's son, also named Bob, served from 2003 until early 2018 as the Milwaukee Brewers' executive vice president, finance and administration, the fourth generation of the Quinn family to work in senior management positions in professional baseball.

Robert E. Quinn grew up in Newton, Massachusetts, and was 17 when he moved with his family to Milwaukee with the Braves in 1953. He graduated from Marquette University in 1958 and began his baseball career in the Braves' and Phillies' minor league organizations. In 1967, he helped establish the Reading Phillies as Philadelphia's Eastern League farm team, winning The Sporting News' Minor League Executive of the Year Award as a first-year general manager there. Reading celebrated its 50th straight year as the Phillies' Double-A affiliate in 2016. Quinn then helped to re-establish a minor league franchise in Omaha in 1969–70 as the first front-office boss of the Omaha Royals of the Triple-A American Association. That franchise celebrated its 50th consecutive season as the Kansas City Royals' top affiliate in 2018.

Quinn joined Milwaukee's current Major League team, the Brewers, as their farm system director in 1971. In 1973, he accepted a similar post with the Cleveland Indians, and spent over a dozen years working as the Indians' scouting and player development boss. He was promoted to vice president in 1981.

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