2 of 4 [Edit removed during grace period]; added 46 characters in body

Seder Olam Rabbah 8 says the Jews left Sinai on 20 Iyar of Year 2 after the Exodus, traveled to Kibrot Hattaavah, spent 30 days there (because that's how long they ate the quail for Num 11:20), traveled to Chatzerot, spent 7 days there (because that's how long Miriam was expelled for Num 12:15), traveled to Midbar Paran, arriving on 28 Sivan or the same year. That's a span of 38/39 days including both end dates to cover 37 days of encampment plus 3 travels (he doesn't mention if Iyar that year had 29 or 30 days).

So the fight probably took place 20, 21 or 22 Iyar.