Within the TV series itself, there's no further mention of the tribbles or their fate.
There are, however a couple of EU sources that may shed some light;
According to the DS9 Comic "Nobody Knows the tribbles I've seen", the maintenance crews collected them all up and then beamed them into space. Quark also claims that Worf went on a mad rampage with a phaser rifle (on the promenade, no less) but nobody believes him.
Within the wider trek EU canon, we see that several of the tribbles escaped from the station and made their way to various Bajoran colonies (and eventually Bajor) in the story from "The Tribbles' Pagh" from Strange New Worlds IX
After threatening to cause another famine in Tozhat province. The
Bajorans plan to kill them with "disposal units" until Doctor Bashir manages to
synthesise a virus that limits their reproductive cycle.
Kira took a deep breath. “Bad. Dax says that there are at least a
hundred thousand tribbles in the Tozhat Province. If they reproduce
unchecked we’re looking at over a million within twenty-four hours.
Trillions within a week.”
“I remember,” Sisko said gravely. “I thought the station quarantine
Kira gave a short barking laugh. “It wasn’t. Shakaar has instituted a
quarantine of Tozhat. We’re trying to track down anyone who entered
the province in the last two days. All shipments of salom grass are
being screened for tribbles. Elimination is a problem. We could use
some industrial disposal units.”