Are you asking for a proof of the "fact" that all proofs of the theorem about cubic surfaces having 27 lines are non-trivial? If so, then no, there is not even a definition of non-trivial proof, let alone a proof that any specific proof is non-trivial or a proof that there exists a fact all of whose proofs are non-trivial.
Maybe years ago that would have been a plausible definition, but by now many proofs have been verified by computer, including several which probably no-one would feel comfortable calling trivial.
One particular proof (of which we do not know if it is exclusive) is non-trivial (whatever that means)? Or, perhaps, all known (to the author) proofs are non-trivial?
Baez didn't mean that as a mathematical statement rather as a statement in the field of history and human psychology: "So far nobody has been able to devise a proof that is widely regarded by people who study it as being trivial (where triviality of a proof is a subjective judgment for which there is (1) no formal criterion, but also (2) a surprising amount of agreement)."
