DS9 showed us the Klingon Chef and the Klingon Lawyer. I don't think those guys fit the usual stereotype.
ENT did a little something like this, too. In the season two episode "Judgment" there was a Klingon lawyer (whose parents were a teacher and a biologist) who lamented that Klingon society used to be about more than just honor and glory in battle. And in the season four episodes "Affliction" and "Divergence" there was a Klingon doctor who was born into the warrior caste but chose to be a healer instead.