A scan at BMI Ross Hall Hospital revealed that Alex sustained a bone injury in the 13th minute of the club’s final regular PRO12 match against Connacht in Galway.
Alex will not require surgery and will now begin a period of rest and recovery with his club's medical team.
Glasgow Warriors head coach Gregor Townsend said: "We're disappointed for Alex that his season has been cut short through injury.
"He had been playing very well and played a big role in getting us into the GUINNESS PRO12 Play-Offs.
"We wish him all the best with his rehab and we look forward to him returning to training ahead of the new season."