We report on an electronic coincidence detection circuit for quantum photonic applications implemented on a field-programmable gate array (FPGA), which records each the time separation between detection events coming from single-photon detectors. We achieve a coincidence window as narrow as 500 ps with a series of optimizations on a readily-available and affordable FPGA development board. Our implementation allows real-time visualization of coincidence measurements for multiple coincidence window widths simultaneously. To demonstrate the advantage of our high-resolution visualization, we certified the generation of polarized entangled photons by collecting data from multiple coincidence windows with minimal accidental counts, obtaining a violation of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality by more than 338 standard deviations. Our results have shown the applicability of our electronic design in the field of quantum information.