The phrase "to fail a quest" means that someone was not successful in completing a specific task or mission that they were trying to accomplish. Full definition
Even after dying without killing the dealer's champion at the end of the dungeon, effectively failing your quest, you can still collect the booster packs for encounters you have completed. (gamingtrend.com)
This is the most common reason why job applicants fail their quest to getting a job. (resumesamples2017.com)
Even when you've just failed a quest because a pink gorilla caught you up against a wall in an unlucky death - combo, you can't help but smile when you're returned to your village and see a little pig wandering around in a sombrero. (kotaku.com)