To trigger the Bubbul Gem quest, search around Pico Pond near Woodland Stable in the Elden Region to meet up with Kilton, a character from The Legend of Zelda: Breath of the Wild. He will be able to direct you to the person you need to meet to trigger The Hunt for Bubbul Gems Quest. And the more Gems you have upon triggering the quest, the closer you are to completing it before it even starts.