Quantum Random Oracle Model
You are right, the success probability of a $q$-query quantum adversary is $\Theta(\frac{k(q+1)^2}{n})$. You can find a proof of the upper and lower bound in this paper.

The upper bound uses similar tools as for the optimality of Grover. The tricky part is that the proof of optimality for Grover's algorithm is given for the worst-case but I guess you and people in crypto in general are mainly interested in the average case complexity. So you need a more advanced toolkit.

The lower bound is more or less applying Grover and analysing its success probability for the average case.