I think what you are looking for is an adversary that
- has access to a quantum computer, and
- is efficient (i.e., runs in polynomial time -> independent of the property it tries to attack).
In this case, the common way to model the adversary is just as a polynomial time quantum algorithm. Note, it depends on the security model for the property that the adversary tries to attack (e.g., collision resistance, or EU-CMA) if it gets quantum access to a primitive or not, not on the adversary model. Schemes which achieve security in terms of traditional security definitions (i.e., with no quantum access to secretly keyed resources) against any polynomial time quantum algorithm are what we call post-quantum cryptography.
As in the traditional / classical setting, the question if a certain problem like the MQ-problem still is hard when considering such polynomial time quantum algorithms boils down to an assumption again: We do not have a proof that there exists no polynomial time quantum algorithm that solves MQ / SVP / [choose your favourite PQ problem] with noticeable success probability. However, we got reason to believe that no such algorithm exists.