Quinn, who tossed just two touchdowns while compiling 1,141 yards in 10 games last season, is expected to serve as backup to Russell Wilson in Seattle.
The 28-year-old became expendable when the Chiefs obtained Alex Smith from the San Francisco 49ers in February.
Quinn also served as a reserve signal-caller for the Denver Broncos for two seasons prior to joining the Chiefs.
A Cleveland Browns draft pick, he has thrown for 3,043 yards with 12 touchdowns and 17 interceptions over six seasons.
The Seahawks this month traded former backup quarterback Matt Flynn to the Oakland Raiders.