A topological frequency converter represents a dynamical counterpart of the integer quantum Hall effect, where a two-level system enacts a quantized time-averaged power transfer between two driving modes of incommensurate frequency. Here, we investigate as to what extent temporal coherence in the quantum dynamics of the two-level system is important for the topological quantization of the converter. To this end, we consider dissipative channels corresponding to spontaneous decay and dephasing in the instantaneous eigenbasis of the Hamiltonian as well as spontaneous decay in a fixed basis. The dissipation is modelled using both a full Lindblad and an effective non-Hermitian (NH) Hamiltonian description. For all three dissipation channels, we find a transition from the unperturbed dynamics to a quantum watchdog effect, which destroys any power transfer in the strong coupling limit. This is striking because the watchdog effect leads to perfectly adiabatic dynamics in the instantaneous eigenbasis, at first glance similar to the unperturbed case. Furthermore, it is found that dephasing immediately leads to an exponential decay of the power transfer in time due to loss of polarization in the mixed quantum state. Finally, we discuss the appearance in the effective NH trajectory description of nonadiabatic processes, which are suppressed in the full Lindblad dynamics.