Bears won the Rose Bowl.
This has yet to happen. Badgers Football won the Rose bowl in 1999 and 2000. The Green Bay Packers won the Superbowl in 1967, 1968, and 1997.
Jim Plunkett was the starting QB for Stanford when they won the Rose Bowl in 1970 and won two Super Bowls as starting QB with the Raiders in 1981 and 1984. Tom Brady was a backup QB for Michigan in 1998 when they won the Rose Bowl and won three Super Bowls as a starter with the Patriots. Brian Griese was the starting quarterback for the Michigan team that won the Rose Bowl and a backup on the Denver Broncos team that won the Super Bowl in 1999. There may be more but that's all I can think of.
Oregon won the 2012 Rose Bowl, beating Wisconsin 45 to 38.
The USC Trojans have won the Rose Bowl the most times with 24 wins.