"unending quest" is a correct and usable phrase in written English.
It typically refers to an ongoing pursuit to acquire something or reach a certain level. For example, "He spent his life in an unending quest for knowledge."
The important thing about science is it's an unending quest".
Family and his many friends will miss his unending quest for knowledge, enthusiasm, energy, and charm.
Unravelling myths, searching for the truth behind the stories, is an unending quest.
The unending quest of historians for understanding the past – that is, 'revisionism' – is what makes history vital and meaningful.
I mean (ominous music, please) the System, which in its unending quest for profit eats up and spits out those trying to hold on to their humanity.
In the unending quest for effective ways of adapting to climate change, it seems that musk ox and caribou may have some of the answers.
But he and his colleagues knew that the human quest for knowledge is unending.
We illustrate how the quest for integration is an unending process and it is produced concurrently and episodically.
If history is written for the present, not the past, then the quest for the historical Bible becomes an unending endeavor subject to the vicissitudes of times, talents and testimonies.
A better comparison might be with New York art dealer Peter Harris from Cunningham's 2010 novel, "By Nightfall," which centers on a character who, like his author, is conscious of "the unending effort to find a balance between sentiment and irony, between beauty and rigor," in the quest to "open a crack in the substance of the world through which mortal truth might shine".
