My searching has only revealed monster “catfish” and trout in the river. I have a lot of new biofortean references I’ve gone through, and not come across anything specific to that region. It would be interesting to know how they came up with the reference for the show.
There does appear to be a legend of a “Nessie”-type critter inhabiting Lake Murray in South Carolina, but I haven’t seen anything about monsters inhabiting the Cumberland River (aside from the aforementioned catfish). See here, for example.