( ͡° ͜ʖ ͡°)
Like, quests with shiny Pokémon for rewards, or about shiny Pokémon
After reading through all the quest text in the game, there is only one that mentions a shiny Pokemon, "A Peculiar Ponyta". You can get the quest around mid-game, I'd say.
But there are other ways to encounter shiny Pokemon such as getting the Shiny Charm and getting mass outbreaks which increase the chances.