We propose a single-shot quantitative differential phase contrast (DPC) method with polarization multiplexing illumination. In the illumination module of our system, the programmable LED array is divided into four quadrants and covered with polarizing films of four different polarization angles. We use a polarization camera with polarizers before the pixels in the imaging module. By matching the polarization angle between the polarizing films over the custom LED array and the polarizers in the camera, two sets of asymmetric illumination acquisition images can be calculated from a single-shot acquisition image. Combined with the phase transfer function, we can calculate the quantitative phase of the sample. We present the design, implementation, and experimental image data demonstrating the ability of our method to obtain quantitative phase images of the phase resolution target, as well as Hela cells.