Entanglement can be considered as a special quantum correlation, but not the only kind. It is allowed to exist non-classical correlations even for a separable quantum system. Here we propose two dissipative schemes for generating a maximally correlated state of two qubits in the absence of quantum entanglement, which was raised by [F. Galve, G. L. Giorgi, and R. Zambrini, Phys. Rev. A 83, 012102 (2011)]. These protocols take full advantages of the interaction between four-level atoms and strongly lossy optical cavities. In the first scenario, we alternatively change the phases of Rabi frequencies of two classical driving fields, while the second proposal introduces a strongly lossy coupled-cavity system. Both schemes can realize all Lindblad terms required by the dissipative dynamics, guaranteeing the maximally quantum dissonant state to be the unique steady state for a certain subspace of system. Moreover, since the target state is a mixed state, the performance of our method is evaluated by the definition of super-fidelity G(ρ1, ρ2), and the strictly numerical simulations indicate that fidelity outstripping 99% of the quantum dissonant state is achievable with the current cavity quantum electrodynamics parameters.