Bridge returns to the team that drafted him in 2015. The 27-year-old spent training camp with the Toronto Argonauts but was released prior to the start of the 2019 season.
The veteran passer started four games for the Saskatchewan Roughriders last year, throwing for 131 yards, one touchdown and three interceptions while 110 yards on the ground.
Bridge also became the first Canadian quarterback to start a playoff game since Gerry Dattilio in 1984.
Montreal finished 5-13 in 2018 and is coming off a Week 1 loss to the Edmonton Eskimos. Antonio Pipkin started the contest, but left with an ankle injury.
Vernon Adams Jr. filled in admirably for Pipkin, completing seven of 10 passes for 134 yards and a touchdown.