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It seems likely the Denebians are familiar with the telepathic humanoids of Deneb IV mentioned in TOS. However, the Denebians possibly lack telepathic capability, or they wouldn't have been so easily tricked by Mudd.

It also seems unlikely the Denebians are related to the Bandi of Deneb IV, or that Deneb V is orbiting the same star as Deneb IV in TNG: "Encounter at Farpoint", since settling the actual star system around Deneb (Alpha Cygni) stated a major leap for the Federation into deep space in this TNG episode, taking place around a hundred years after "I, Mudd".

Paragraph 1: No clear reason is given why "it seems likely", and the rest of the paragraph is pointless speculation.
Paragraph 2: Given that we're talking about two differently named species hailing from two different planets, there's no good reason for draging up the notorious "two Deneb IV's" issue on this page -- Capricorn 20:46, June 15, 2011 (UTC)