Classical plasmonics has mostly focused on structures characterized by large dimension, for which the quantummechanical effects have nearly no impact. However, recent advances in technology, especially on miniaturized plasmonics devices at nanoscale, have made it possible to imagine experimental applications of plasmons where the quantum nature of free charge carriers play an important role. Therefore, it is necessary to use quantum mechanics to model the transport of charge carriers in solid state plasma nanostructures. Here, a non-local quantum model of permittivity is presented by applying the Wigner equation with collision term in the kinetic theory of solid state plasmas where the dominant electron scattering mechanism is the electron-lattice collisions. The surface plasmon resonance of ultra-small nanoparticles is investigated using this non-local quantum permittivity and its dispersion relation is obtained. The successful application of this theory in ultra-small plasmonics structures such as surface plasmon polariton waveguides, doped semiconductors, graphene, the metamaterials composed of alternating layers of metal and dielectric, and the quantum droplets is anticipated.