The injury means the British & Irish Lion will miss the first two rounds of the Amlin Challenge Cup against Top 14 side Bayonne and Italian outfit I Cavalieri Estra.
Monye is scheduled to return to action in December but faces a fitness race ahead of that month's double-header against Connacht.
Quins first welcome Connacht to the Stoop on December 12 before crossing the Irish Sea to take on the Galway-based province at the Sportsground at week later.
The Twickenham based club will be hoping to repeat the successes of the 2000/01 and 2003/04 seasons when they lifted the European Challenge Cup.
A statement on the Quins website claimed the surgery was a success and said: "The extent and duration of the injury will be known more clearly after further consultation with the surgeon, and further updates will be given."