A monk is a member of a religious order, typically an individual who takes on vows and lives their life to rules decreed by their religion.
The Klingon monks at the monastery on Boreth were considered to be the most devoted followers of Kahless the Unforgettable in the 23rd century. (DSC episode: "Point of Light"). In the 24th century, seeing the decaying state of Klingon society a group of these monks created a clone of Kahless. (TNG episode: "Rightful Heir")
In 2369, an away team from the USS Enterprise-D encountered a group of monks from a Dracon monastery on the planet Riat. These monks had been poisoned by algae in their water supply which made them cannibalistic and more aggressive than usual. The monks attacked the away team, who only just escaped. Once the away team were rescued, Doctor Beverly Crusher was able to cure the monks of their ailment. (TNG comic: "Light of the Day")