We propose a quantum sensor network for direct detection of quantum chromodynamics (QCD) axion dark matter using entangled superconducting qubits. In the presence of a static magnetic field, the oscillating axion field induces a coherent qubit rotation described by an encoding angle proportional to the axion-photon coupling and an unknown dark-matter phase. Treating the phase as an additional unknown parameter turns axion detection into a two-parameter quantum estimation problem. We formulate these signals in Cartesian coordinates, thereby avoiding the singularity that arises in the weak-signal regime and establishing a regular framework for multiparameter quantum estimation. Using this framework, we optimize the quantum sensor network and combine it with Bayesian inference to reconstruct the axion-photon coupling. The resulting protocol accurately reconstructs the underlying axion-photon coupling despite the unknown dark-matter phase, recovering the KSVZ and DFSZ benchmark models across the target mass range $m_a\in[0.1,10]~\mu{\rm eV}$, generalizing to previously unseen masses, and remaining robust under realistic superconducting-qubit decoherence and gate errors.