We give the first deterministic algorithm that makes sub-quadratic queries to find the global min-cut of a simple graph in the cut query model. Given an $n$-vertex graph $G$, our algorithm makes $\widetilde{O}(n^{5/3})$ queries to compute the global min-cut in $G$. As a key ingredient, we also show an algorithm for finding $s$-$t$ max-flows of size $\widetilde{O}(n)$ in $\widetilde{O}(n^{5/3})$ queries. We also show efficient cut-query implementations of versions of expander decomposition and isolating cuts, which may be of independent interest.