It means Monye, named as a member of the England Saxons squad in the summer, will miss the Twickenham internationals against Australia, New Zealand, South Africa and Samoa.
Monye, who has won 13 caps in the Red Rose jersey, suffered the injury in Quins' 29-29 draw against Wasps on the opening day of the season.
But he was absent as his team mates slipped to a 21-17 defeat against Sale Sharks on Friday night.
Elsewhere, 2010 Aviva Premiership runners-up Saracens cruised to a 26-14 victory at Leeds with England hopeful David Strettle on the scoresheet.
Meanwhile, fly-half Ryan Lamb kicked 13 points for London Irish as they won 23-16 against his former club Gloucester.
"You're going to get a bit of x-factor with Ryan but what has impressed us this season has been his kicking consistency," said Irish coach Toby Booth.