We study consensus seeking of quantum networks under directed balanced interactions defined by a set of permutation operators among a network of qubits. The state evolution of the quantum network is described by a continuous-time master equation. We show that the network state always converges along the equation, and the convergence limit is determined by the generating subgroup of the permutation operators. We also give a tight graphical criterion regarding when such limit admits a reduced-state consensus. Quantum consensus conditions under switching interactions are also established with the help of cut-balanced consensus processes in the classical setting.