The right-back tested positive for clostebol - which is prohibited by the World Anti-Doping Agency - following a World Cup qualifying match on February 2.
"The sanction followed the player's admission of the violation and the subsequent conclusion of a case resolution agreement between the player, FIFA and WADA," FIFA said in a statement.
"The sanction is therefore final and binding."
FIFA said that since the period of provisional suspension served by Quaye had been credited against his 18-month period of ineligibility, the player was banned until August 1, 2023.