Surface plasmon polaritons (plasmons) are a combination of light and a collective oscillation of the free electron plasma at metal-dielectric interfaces. This interaction allows sub-wavelength confinement of light, beyond the diffraction limit inherent to dielectric structures. The resulting electromagnetic fields are more intense and the strength of optical interactions between metallic structures and light-sources or detectors can be increased. Plasmons maintain non-classical photon statistics and preserve entanglement on plasmon-assisted transmission through thin, patterned metallic films or weakly confining waveguides. For quantum applications it is essential that plasmons behave as indistinguishable quantum particles. Here we report on a quantum interference experiment in a nanoscale plasmonic circuit consisting of an on-chip plasmon beam splitter with integrated superconducting single-photon detectors to allow efficient single plasmon detection. We demonstrate quantum mechanical interaction between pairs of indistinguishable plasmons by observing Hong-Ou-Mandel interference, a hallmark non-classical effect which is the basis of linear optics-based quantum computation. Our work shows that it is feasible to shrink quantum optical experiments to the nanoscale and demonstrates a promising route for sub-wavelength quantum optical networks.