For the psychological measurement, please see Duke-Heidelburg quotient.
Duke (male) or duchess (female) was a title of nobility used, among other places, on Renavi.
In the late 2250s, Harcourt Fenton Mudd was wanted by the United Federation of Planets for "attempted regicide," attempting to assassinate a duke, though Mudd didn't feel a duke should qualify as "regicide." (ST: "The Escape Artist")
In 2266, Mudd promised Ruth Bonaventure that upon wedding a lithium miner she'd be (as rich as) a duchess. (TOS: "Mudd's Women")
Kor and Curzon Dax broke into the duke's harem on Renavi. Years later in 2375, Kor asked Ezri Dax if she remembered that event, and she laughed, stating "How could I forget? It's one of Curzon's favorite memories..." (DS9: "Once More Unto the Breach")