Unfortunately, there is no mention anywhere else in Tokien's oeuvre of when Strider's earlier adventures in Moria occurred, nor what they amounted to. I too wondered about this question, but (unless I missed it) there was no information about it to be found it, neither in the author's completed works, nor in the notes published in The History of Middle Earth volumes.
This blog post from a few years ago comes to the same conclusion about the matter, although the author hazards some guesses about when Aragorn's sojourn in Khazad-dûm might might have occurred:
The most popular choice seems to be the years 2980 to 2989, after he finished his errantries in Rohan and Gondor and before Balin arrived in Moria.
Although when I first read Aragorn’s statement many years ago I assumed that he had passed all the way from the Dimrill Gate to the West-gate, and exited Moria by pushing the doors open, I have changed my mind. I think Aragorn’s first journey into Moria began and ended near the Dimrill Gate. He displays no knowledge of the west-gate’s operation when the Fellowship stands outside and offers no help in guiding the party through Moria. Gandalf and Gimli handle that task. I think Aragorn was as unfamiliar with the western side of Moria’s tunnels as everyone else but Gandalf.
He was also unaware of the presence of the Balrog (for he surely would have said something had he known such a monster was there). And he knew nothing of the fate of Balin’s colony
Aragorn’s journey with Gollum from Mordor to northern Mirkwood occurred in 3018 and the narrative says he explicitly avoided Moria and Dol Guldur, so he must already have known there were perils in Moria. Nor did he seem to know that Gollum had escaped from the Elves, so he could not have chased Gollum into Moria.