The winger leaves the club he signed for from Inter in 2010 after accepting a cancellation of his current deal with the Black Eagles.
The termination was amicable and was mutually accepted by both parties, making the 29-year-old available for a free transfer.
It is understood the Turkish side has paid the winger for the remaining four months on his current deal that was due to expire in the spring.
Quaresma had struggled to adjust to life at Besiktas and was reportedly willing to sue the club in October, and told the media that the club did not want him.
He was brought to the side by Bernd Schuster from Inter in 2010.