But left-back Kenny van der Weg and both Chris Burke and Tony Dingwall, who are recuperating from fitness issues, are in talks over contract extensions.
The club also confirmed the departure of Jonathan Franks and Oscar Gobern.
Former Motherwell, Cardiff and Aberdeen defender Quinn, 31, has had several lay-offs with hamstring injuries in recent months and made 22 appearances this season. Franks had been told he was free to leave in January while Gobern only made one appearance after joining from Mansfield in January.
Woods played 35 times this season and has triggered a one-year extension while former Queen of the South midfielder McShane made 11 appearances and the pair have been told they can leave on free transfers.