We propose a quantum plasmonic sensor using Hong-Ou-Mandel (HOM) interferometry that measures the refractive index of an analyte, embedded in a plasmonic beam splitter composed of a dual-Kretschmann configuration, which serves as a frustrated total internal reflection beamsplitter. The sensing performance of the HOM interferometry, combined with single-photon detectors, is evaluated through Fisher information for estimation of the refractive index of the analyte. This is subsequently compared with the classical benchmark that considers the injection of a coherent state of light into the plasmonic beamsplitter. By varying the wavelength of the single photons and the refractive index of the analyte, we identify a wide range where a 50 % quantum enhancement is achieved and discuss the observed behaviors in comparison with the classical benchmark. We expect this study to provide a useful insight into the advancement of quantum-enhanced sensing technologies, with direct implications for a wide range of nanophotonic beamsplitter structures.