Alexander, who joined the Charges midway through last season having previously played for the St Louis Rams, suffered a torn right anterior cruciate ligament (ACL) during Tuesday's practice session.
The 25-year-old impressed for the Chargers with 37 passes for 658 yards and seven touchdowns.
San Diego missed out on the play-offs last season with the Chargers finishing with a 7-9 record in the AFC West behind Divisional winners the Denver Broncos.
It proved a bad day to practice for wide receivers on Tuesday with the Eagles' Benn also tearing his ACL.
The former Tampa Bay Buccaneer, who has five touchdowns in his professional career, was serving as a blocker on a kick-off drill when he suffered the injury.