The phrase 'on their quest' is grammatically and stylistically correct and can be used in written English.
For example: "The brave adventurers set out on their quest to defeat the evil sorceress."
Remember in book seven when Ron was the most affected by the locket horcrux and pussed out, momentarily ditching Harry and Hermione on their quest to destroy Voldemort?
The great magician Merlin helps them on their quest.
The bots are on their quest to deliver lunch.
The investigators conceded that much is riding on their quest.
Despite the setbacks, researchers say they cannot give up on their quest for cancer signatures.
Each month they check in and spin comedy based on their quest.
One idea is a faux documentary that follows Team Rocket on their quest of crime.
It's the continued late-game collapses that are hurting LA on their quest to get a third straight MLS Cup.
College students can embark on their quest for environmental improvement by learning about the work of experts like Jordan Rubin.
On their quest to restore hearing through gene therapy, scientists have long sought ways to improve gene delivery into hair cells.
A good travel book is a treasury of wisdom that seeps into your blood as you follow the author on their quest.
