Kay has been carrying the problem in recent weeks but the club took the decision to operate and he went under the knife on Sunday afternoon.
The 30-year-old had played in all but one of the Dons' games this season.
Head of sports medicine Simon Crampton told the club's oficial website: "Antony has had a hip problem over the last month; investigations had revealed this was due to a tear of the labarum in the hip joint.
"It had been our plan to manage this in the short term which meant that although Antony had been able to play the last four games, we had to modify his training throughout the week.
"Obviously this was not an ideal situation and Antony was playing through the pain barrier to get through games and therefore the decision was taken for him to have surgery over the weekend.
"The surgery was successful and he will now undergo a rehabilitation period but unfortunately it means he will be out for approximately 12 weeks."