Pairs of polarization-entangled photons are important for diverse quantum technologies, such as quantum communication, computation, and imaging. However, generating complex polarization-entangled states has long been constrained by the available nonlinear susceptibility tensor of natural materials, necessitating cumbersome setups for additional coherent superposition or postselection. In this study, we experimentally demonstrate the generation of pairs of polarization-entangled photons using a plasmonic metasurface strongly coupled to an epsilon-near-zero (ENZ) material. By engineering a resonance at the pump wavelength and leveraging the field enhancement provided by the ENZ effect, the photon pair generation efficiency of the 68-nanometer-thick metasurface is substantially boosted compared to that of an unpatterned indium tin oxide film. More notably, the ENZ metasurface platform facilitates versatile manipulation of the system's anisotropic second-order nonlinear susceptibility tensor, enabling direct control over the polarization states of the photon pairs and generating a polarization-entangled Bell state without additional components. This approach enables simultaneous photon pair generation and quantum state engineering in a compact platform.