Open quantum systems far from thermal equilibrium can exhibit remarkable physical phenomena including topological properties without a direct equilibrium counterpart. Along these lines, in periodically driven-dissipative systems within the effective non-Hermitian (NH) Hamiltonian approximation spectral winding numbers have been linked to intriguing nonreciprocal transport properties. Here, going beyond an NH Hamiltonian description, we introduce and study a microscopic lattice model of a driven open quantum system described by a Markovian quantum master equation, which exhibits the mentioned spectral winding within an NH approximation. By encompassing quantum jump processes in the topological analysis, we uncover a distinct jump-induced topological phase, which qualitatively corresponds to the richer nonreciprocal transport properties of the fully quantum model. In addition, we find that the NH skin effect, i.e., the accumulation of a macroscopic number of eigenstates at one end of the system, is already visible in the transient dynamics even for systems with periodic boundary conditions. Our results exemplify the subtle correspondence between NH topological properties and physical manifestations of Liouvillian topological properties in open quantum systems, thus providing a theoretical framework toward understanding unidirectional transport in quantum dissipative Floquet dynamics.