Yes, there is a bike route from the Queensboro Bridge.
It goes down Sutton Place to 53rd St
Take 53rd St. to 2nd Ave
Take 2nd Ave down to 38th St
Take 38th St to the East River Path
Take the East River Path down to 20th St
Take 20th St to Park Ave South
Take Park Ave South to Union Square
Obviously you can go anyway you want, but this route keeps you on Bike Lanes/Bike Routes/Bike Paths for the entire trip except the last little stretch on Park Ave South