'DS9:Trials And Tribble-ations' seems to be implying this.
WORF: They were once considered mortal enemies of the Klingon Empire.
ODO: This? A mortal enemy of the Empire?
WORF: They were an ecological menace, a plague to be wiped out.
ODO: Wiped out? What are you saying?
WORF: Hundreds of warriors were sent to track them down throughout the galaxy. A armada obliterated the Tribbles' homeworld. By the end of the twenty third century they had been eradicated.
ODO: Another glorious chapter of Klingon history. Tell me, do they still sing songs of the great tribble hunt?
But I've been having a look on Ex Astis Scientia, and it's clear that the Tribbles still existed as late as 'Generations' because we actually see a child holding one (and if we take Keiko's school-room board as read, then they were at least known
in the 24th century).
(I actually think Alexandra's "pet" in 'TNG:When The Bough Breaks' is only supposed to be a plush Tribble, if it's a Tribble at all. Sure it's got a cage, but that might just be an accessory that the plushy came with. This is supported by the scene where she cheekily sticks it to Picard's back.
Maybe the one in 'Generations' was only a plushy as well?)
Any answers for this? Are we to assume that Worf's line isn't that literally
the Tribbles were eradicated, but simply that their homeworld was? And therefore the only Tribbles left were a scattered number 'out in the wild'? (Making them only an endangered species in the 24th century, rather than an extinct one?)
One thing that does need to be noted about 'DS9:Trials And Tribble-ations' is that if we watch it carefully we can see that none of the Starfleet officers seem to react with curiosity about the Tribbles at all. Even Bashir acts like he knows what a Tribble is
when he first sees one, but he just isn't expecting to see it in the Enterprise corridor. The only DS9 character who overtly
exhibits no knowledge about them in the episode is Odo, which is entirely excusable because it's very unlikely he's ever seen one before.
Of course, if Tribbles are still knocking around in the 24th century in some fashion, it does rather destroy the punchline of 'Trials And Tribble-ations' (ie, that Sisko and crew have inadvertently reintroduced them to the galaxy).