Afraid not. The logical time would be during his 'great journeys' prior to the War of the Ring, but nothing specific is ever said. Why? Who knows. Perhaps he also was looking for Thrain- but then it would be odd that he and Gandalf went separately.
I would rule out the search for Gollum- G. doesn't appear to have made it as far as Moria before Aragorn caught him, and T. writes that when G did enter Moria after his escape from Thranduil he got lost and trapped.
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