The former Scotland Under-21 internationalist is understood to have made his feelings clear towards Doonhamers supporters as he left the pitch at half-time as Gary Naysmith’s side slipped to a seventh straight loss to Inverness on March 9.
Martin was replaced by Jack Leighfield at the interval and the club have taken action after completing their own investigation.
A club statement read: “Following an internal investigation into events surrounding the incident involving Alan Martin during our game against Inverness Caledonian Thistle, we can confirm that a one match suspension and a fine was imposed by the club.
“The club are now satisfied that this matter has been dealt with and would like to draw a line under it.
“We need to stick together and back the team for the remainder of the season.”
Martin, who would have sat out the weekend’s clash with Ayr United had it not been postponed, admits he regrets his actions.
He said: “I would like to publicly apologies to everyone concerned, the supporters, the directors, the manager and my team-mates.
“I know what I did was unacceptable and it shouldn’t have happened. I fully accept the punishment the club have imposed.
“To clarify, I had been receiving personal abuse during the pre-match warm up, during the game and at half-time, and frustration got the better of me. But there is no excuse for what I did and I shouldn’t have reacted.
“I would have apologised before now, but felt that it wasn’t appropriate ahead of such a big game.
“Our full focus had to be on the match and I wanted to give Jack my full support and make sure that we were fully prepared for Ayr.”