偶然発見するを英語でなんと言うか？

chance upon です。0:56 two weeks ago, amateur divers chanced upon a number of coins, which they at first thought were children's toys. but soon realized what they had found.

Huge hoard of gold coins from 11th Century unearthed off Israeli coast
https://www.youtube.com/watch?v=pSPQ24y-bnI  0:04 in 2002 Romanian cavers chanced upon a new chamber deep inside the cave.
FIRST PEOPLES | The First Modern Human | PBS