Is this a trick question? Jim Plunkett won in 1981 with the Oakland Raiders and again in 1984 with the LA Raiders.
No quarterback has won the Super Bowl as a starter on two different teams.
No starting quarterback has won a Super Bowl ring with two different teams.
The quarterback that was on the most Super Bowl teams was backup Gale Gilbert. He was on the roster of the four Buffalo Bills teams that went to the Super Bowl (Super Bowls XXV through XXVIII) and the San Diego Chargers team that played in Super Bowl XXIX, a total of five Super Bowls. Incredibly, his team lost the Super Bowl each time.
Tom Brady and Ben Rothlisberger
Ken Norton, Jr.
No quarterback has won the Super Bowl with different teams. The only starting quarterbacks to take two separate teams to the Super Bowl are Craig Morton and Kurt Warner. Morton led the 1970 Dallas Cowboys to the big game, but lost to the Baltimore Colts 16-13. He later led the 1977 Denver Broncos to the Super Bowl, but lost the game to the Dallas Cowboys 27-10. Warner took the Rams to Super Bowls XXXIV and XXXVI, winning one and losing one, and the Cardinals to Super Bowl XLIII, losing to the Steelers.