Are the words 'renege' and 'renegotiate' related, etymologically?

They are not directly related.

Both contain an element “ne-” meaning “no/not” in there, but they are from separate Latin roots. “negotiate” (and thus also “renegotiate”) is from “nec otium”, literally “not leisure”, i.e. “business”. “renege” is from the verb “renego”, which is a variant of “nego”, meaning to “deny” and is simply the word for “not” turned into a verb.

Wiktionary makes the note that “renege” may have been influenced by “renegotiate”. There is an older Middle English word “renay” which meant the same thing, “to renounce, to go back on”.

Only that they have Latin origins.
Per AI
“Renege” means to break an agreement, deal, or promise.
“Renegotiate” means to discuss an agreement again to change it, especially to alter or change previously agreed terms.