Some people think the use of "cod" for phoney derives from an "old codger", but the term is actually seen earlier than the use of the slang term "codger". It relates probably to cod-head, like a fish's head looking stupid, and therefore the etymology of the cod fish itself, which is cognate with the Latin Gadus (which you see in its Linnaean name) and in fact the only one of the modern European languages which does actually derive from Gadus, is the most likely etymology in my opinion. Other European languages have terms like Bacallau, Kabeljau, Dorsch or the related Torsk, thorsk, dorsz, or they have terms deriving from morhua, like French and Italian have.

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Cod-piece also exists in English, but sources reckon that is not related, and comes from a lost Germanic term for a pouch.