The Ghana international had joined Pompey on a season-long loan from Spartak Moscow in January, however, the deal has been ended prematurely after the Russian club accepted a fee of around £3million.
Pompey reportedly paid an up-front fee of £500,000 for the 23-year-old's services and Spartak are set to pay half of that sum back to the crisis-hit club in compensation for ending the deal early.
Quincy flew to the Middle East on Tuesday and has signed a three-year deal to join the 11-time Qatar champions.
"I'm happy with the good reception from the administration and fans of Al Sadd," Quincy told the club's website.
"I have a good feeling and this will give me a big push to have good performances with the team.
"I came to Qatar in order to give all what I have to Al Sadd and help them to win a lot of matches and titles as well.
"I will enjoy playing with Al Sadd because they are one of the best teams in the region."