The collective noun most often used for a group of finches is a charm. You may also hear a few other terms for them, including a trembling, a trimming, and a chirm. The word charm is believed to trace back to the Old French charme, which referred to a song or a flock of birds. That link makes sense, since finches are well known for their tuneful singing.
What Is a Group of Finches Called? – Finches Collective Nouns
