The 27-year-old suffered the injury during the surprise 3-0 defeat at Hereford 10 days ago and went under the knife on Monday.
The club, who can clinch promotion from League Two this weekend, have also said that Robertson's recovery could see him miss the start of the 2011/12 campaign in August.
Robertson played 14 matches last season before he suffered a broken leg in November against Carlisle in the Johnstone's Paint Trophy.
It took the Scot a year to recover from that injury and he made his return to first-team action at Harrow in the FA Cup.
He played another 14 times this season before he broke down at Edgar Street.
Source: PA
Source: PA