“Chaucer when warning against envy in the in the Parson’s Tale uses the word <murmuration> ‘After bakbitynge cometh grucchynge or murmuracioun..’ (499) in modern English: After backbiting comes grutching or murmurance [grumbling & complaining].
From this source we know the word has been attested in English as far back as 1390. We see in the OED it was used in reference to starlings in 1450:’ in PMLA (1936) 51 603 (MED) ‘A murmuracione of stares’. We analyzed this as <murmur+ate+ion> and as Jin noted above, the base element entered English via Old French meaning ‘sound of human voices, trouble or argument’ from Latin murmurare which has origins in a reduplicated PIE root*mor-mor meaning hum, muttering rushing.
Jin commented on the Greek cognate mormyrien : meaning to boil. When you look at the murmuration clip below you can hear the rushing and see the ‘boiling’ hum of starlings.”
https://www.youtube.com/watch?v=KxfvseECDcs