We demonstrate experimentally the ability of a quantum annealer to distinguish between sets of non-isomorphic graphs that share the same classical Ising spectrum. Utilizing the pause-and-quench features recently introduced into D-Wave quantum annealing processors, which allow the user to probe the quantum Hamiltonian realized in the middle of an anneal, we show that obtaining thermal averages of diagonal observables of 'classically indistinguishable' non-isomorphic graphs encoded into transverse-field Ising Hamiltonians enable their discrimination. We discuss the significance of our results in the context of the graph isomorphism problem.