Nope. Per is related to a whole family of Indo-European prepositions and prefixes, including para-, peri-, pro-, for-, and fore-. By itself the preposition is basically just equivalent to for.
Person is obscure but may have been borrowed into Latin from the Etruscan phersu, which means "mask". It then came to mean "character in a play" (i.e., someone who wears a mask) before broadening to its current meaning. It was then borrowed into English.