The injury is almost identical to the one Azam suffered to his right shoulder last season and is likely to keep the Frenchman out of action until the end of February.
Azam suffered the setback in October but appeared to be close to a potential return when he resumed contact training before Gloucester's Guinness Premiership trip to London Irish last month.
However, the injury, which affects the stability of the shoulder, did not settle down and required a subsequent operation.
It leaves Wales international Mefin Davies as Gloucester's senior hooker, with Rob Elloway and James Parkes also in contention.
Gloucester's head coach Dean Ryan told the club's official website, www.gloucesterrugby.co.uk: "Everyone at the club is obviously very disappointed for Olivier, particularly after the problems he suffered last year.
"To sustain another serious injury so quickly is disappointing for him and to us because we will miss the huge contribution he makes to the team.
"All we can do now is wish him a speedy recovery so he is back with us as soon as possible."