The involvement of the quantum computer is only that it's an adversary that can break asymmetric encryption with different complexity constraints than a classical computer. For example, take two random prime numbers as a secret and publish the result of multiplying them. It turns out that if you want to find the two random prime numbers by knowing only the result of multiplication is a hard problem known as integer factorization. If you double the size of the prime numbers, discovering them takes exponentially longer. The theoretical quantum computer model though says that it can accomplish it in sub-exponential time. Now doubling the size of the number only increases your compute requirements by ~2x to recover the prime factors (specifically it's actually a logarithm so it's < 2x more compute is needed). The algorithm for doing this is known as Shor's algorithm [0] and because of how complexity works in computer science, it turns out that this algorithm can be applied to many many problems mechanically and these problems are the underpinnings of RSA, DSA, ECDH key exchanges.
These quantum-resistant algorithms are based on mathematical problems believed to be exponentially difficult even for a theoretical quantum computer - when you double the size of the problem, it again takes exponentially more time even for a theoretical quantum computer. These are of course unproven beliefs but that's true of classical algorithms too. So no, it doesn't matter where you run the cryptographic algorithm; it remains computationally difficult to solve the problem without knowing the secret. The quantum computer is critically important though for your ability to crack classical problems though - without it, all of this post-quantum cryptography is unnecessary.
[0] https://en.wikipedia.org/wiki/Shor%27s_algorithm
[1] https://arxiv.org/pdf/2212.12372