[0001] Regarding the OP's first paragraph and the comments therein, there is no protocol, call it $X$, that can be executed efficiently on classical computers, that has been proven to be secure against quantum computers. If we had such a proof then we would also know that P$\ne$NP. This follows because, as @Martin said, a quantum computer can efficiently simulate a classical computer. If we could prove $X$ to be secure against quantum computers, then it follows that $X$ is secure against classical computers. But if $X$ is efficient to execute, then it follows that $X$ is in NP, since verifying a solution must be efficient. If $X$ is further secure against classical computers, then $X$ would not be in P, which means that we have split P from NP and shown them not to be equal.
[0002] Nonetheless, as @gIS provides and as I think is the real intent of the question, there is a whole field called "post-quantum cryptography" - see the Wikipedia link (mentioned by gIS) as well as the links on a sister site, for example. Herein, although there may be no proof that such protocols are secure against a quantum computer, there is other evidence that such protocols may exist. There is an expectation that we may convert to such protocols if and when RSA is broken by a quantum computer capable of running large enough instances of Shor's algorithm. Many leading contenders for such post-quantum cryptography are based on lattices - the learning with errors class of problems being an example therein.
[0003] Regarding the OP's second paragraph, however, the relationship between what is efficient to execute on a quantum computer (researchers call the class of such problems BQP), and what is efficient to verify on a classical computer (as the OP correctly identified, researchers call the class of these problems NP), is really very rich and dynamic. Indeed, there are lots of reasons to believe that the class BQP and the class NP are incomparable - that is, there are problems (such as those in knot theory) that are easy to solve on a quantum computer, and are not even efficiently verifiable on a classical computer. There are interesting approaches to classical verification of quantum computation for such problems. See, e.g., this Quanta article on a breakthrough of Mahadev, who showed how a classically constrained verifier can force a quantum prover to tell the truth.
[0004] However, post-quantum cryptography for classically secure communication, as discussed above, is not based on or invalidated by the existence of problems in BQP but not in NP - which are precisely the kinds of problems Mahadev's protocol addresses. This is not to say that Mahadev's protocol doesn't rely on post-quantum cryptography (indeed it does), but rather, nothing is precluding us to adopt a post-quantum secure protocol even in the event that we never have cryptologically viable quantum computers.
