Topological quantum interference emerges from the interplay between quantum mechanics and topology. We present evidence for two types of such interference phenomenon that can result from the quantum dynamics of initial topological states. We realize both types of topological quantum interference in a pumped non-Hermitian Su-Schrieffer-Heeger lattice that can be implemented by creation and coherent control of excitonic states of trapped neutral atoms. On quenching the system from the topological to the gapless phases and then back again, we find that interference patterns develop in the gapless phase and also after switching back to the topological phase. These patterns occur both as many-excitation interferences generated in the presence of pumping the atoms at the end sites, and as one- and two-excitation interferences seen in the absence of pumping when starting with edge excitations. Investigation of the excitation dynamics shows that these interference patterns originate from the topological nature of the initial states and are very different from quantum interferences originating from nontopological states of the lattice. Our results also reveal that unlike well-known situations where topological states are protected against local perturbations, in the non-Hermitian SSH systems resulting from driving the excited-state populations, a local dissipation at each lattice site can suppress both the topological interference and the total population of the lattice.