The phrase "quest for" is correct and usable in written English. You can use it to refer to a mission or journey of seeking or searching for something. Example: "The knight embarked on a perilous quest for the Holy Grail.".
A comical quest for a mythical grail.
It meant abandoning my quest for justice".
The quest for real accountability must continue.
Anyhow, their quest for worldwide fame continues.
The quest for comfort knows no borders.
Drag is about the quest for transformation.
It was a quest for "justice".
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union