Alexander, who has scored 12 goals for the Bees this season, has undergone a minor operation on a knee injury.
"The specialist decided to go inside and found some cartilage that needed trimming up," revealed manager Nicky Forster, who is facing up to an injury crisis for the final two matches of the campaign.
Goalkeeper Richard Lee, centre-half Pim Balkestein and forward Charlie MacDonald will also miss the run-in while defenders Craig Woodman and Leon Legge are struggling with injuries.
And centre-back Karleigh Osborne was taken to hospital after he blocked a powerful shot from Notts County's Craig Westcarr with his face during Monday's 1-1 draw.
To add insult to injury, literally, County were awarded a penalty for handball - but Bees goalkeeper Simon Moore saved the spot-kick.