Over the summer Alves made it clear he wanted to leave the club and went AWOL during pre-season, failing to turn up for training.
He later returned but has never been in contention to play in the Championship for Boro.
The 28-year-old has now signed a three-year deal with Al Sadd, who have paid around £6million for the player.
Alves is the fourth big-name departure from The Riverside this season, following the sales of Stewart Downing, Tuncay Sanli and Robert Huth.