Optics is renowned for displaying quantum phenomena. Indeed, studies of emission and absorption lines, the photoelectric effect and blackbody radiation helped to build the foundations of quantum mechanics. Nevertheless, it came as a surprise that the visible transparency of suspended graphene is determined solely by the fine structure constant, as this kind of universality had been previously reserved only for quantized resistance and flux quanta in superconductors. Here we describe a plasmonic system in which relative optical transparency is determined solely by the fine structure constant. The system consists of a regular array of gold nanoparticles fabricated on a thin metallic sublayer. We show that its relative transparency can be quantized in the near-infrared, which we attribute to the quantized contact resistance between the nanoparticles and the metallic sublayer. Our results open new possibilities in the exploration of universal dynamic conductance in plasmonic nanooptics.