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In the canon Prime universe (which is to say from The Original Series to Voyager and not including any alternate timelines) the first time humans (Starfleet or the Federation) encountered the Borg was in 2152 in the Enterprise episode Regeneration. These were drones recovered from the wreckage of the Sphere the Enterprise E destroyed over Earth in Star Trek: First Contact. They were not seen again until the The Next Generation episode QWho in Season 2, because it took so long for the Borg to travel from the Delta quadrant in the 21st Century to the Alpha quadrant in the 24th Century. So no, Kirk never faced the Borg and therefore never defeated them. Whether there is any non-canon story in which the two meet, I don't know, but as it would be non-canon, it wouldn't count.

It is worth noting that I have heard several online rumours about the upcoming, as yet untitled Star Trek film (a sequel to 2009's "Star Trek") that it will in fact include the Borg. Whether this is hearsay, lies or has some amount of truth I don't know.

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