Trebizond is derived from the Ancient Greek Trapezous (genitive Trapezountos, hence Modern Greek Trapezounda), meaning ‘table-like’, and referring to the mountain formation in the area.
There are two theories on how the river got its name. The first one says that it was named “Trebižat” because it escapes from the surface three times. According to the second theory, the name comes from the Italian “il trebizatto”, meaning the river is rich in eels.
So, unlikely, and I wouldn’t have thought there’s a clear reason why you’d name the river after such a far-off city anyway. Especially when the river gets a different name each of the nine times it resurfaces above ground to begin with (Trebižat (river) – Wikipedia)—so none of the originally named instances were particularly long or, I’d have thought, widely known.