A scan on Tuesday showed Max suffered a medial ligament injury and cartilage damage following a tackle just after half-time of Sunday's 34-23 Magners League defeat at the Ospreys.
He was taken off on a stretcher and will now undergo surgery.
Glasgow team doctor Gerry Haggerty said: "The outcome of yesterday's scan shows that Max's anterior cruciate ligament (ACL) is intact, which is positive news.
"He has a medial ligament injury and some cartilage damage.
"He will undergo an operation to repair the cartilage and this procedure will also uncover any injury to the ACL that the scan may not have revealed.
"At this stage, we can estimate that Max could be out for approximately four to six months."
Meanwhile, Warriors head coach Sean Lineen will use all the time allowed to him before announcing his team to face Connacht at Firhill on Friday.
The deadline for submitting a final match-22 is 12noon on Thursday for a Friday game.