The Scotland midfielder, who was named man of the match, opened the scoring early on and further goals from Joe Ledley and Giorgios Samaras saw the champions comfortably home.
With the Hoops already in the last 16 of the Champions League and hitting top gear on the domestic front, Brown was asked if now would be the ideal time to have surgery.
But after enjoying some banter with the assembled press at Rugby Park, Brown became noticeably riled at the frequent questioning of his well-being.
"I think we have had enough talk of this surgery," he said. "It is boring me now, so I think we will get on to other things and look forward to the next game.
"We have a big game on Wednesday against Arbroath and that's the main thing for us now."
Celtic boss Neil Lennon praised the contribution of his captain who looked back to his marauding best at Rugby Park.
"I thought Scott had a great game, he looked brilliant," said the Irishman.
"He is a quality player. His discipline and maturity has improved. Since the last injection he had he has looked more and more like himself.
"The last injection was around two and half weeks ago and it usually takes 10 days to see the full effect.
"He is making the call - obviously the physio as well, but we trust his judgement.
"We know it is not ideal. It's not ideal for him to have surgery either as we don't know if that is going to really resolve the problem.
"We might have the chance to give him a rest but if he is playing like that I am not going to rest him."