We investigate a quantum nonrelativistic system describing the interaction of two particles with spin 1 2 and spin 0, respectively. We assume that the Hamiltonian is rotationally invariant and parity conserving and identify all such systems which allow additional integrals of motion that are second order matrix polynomials in the momenta. These integrals are assumed to be scalars, pseudoscalars, vectors or axial vectors. Among the superintegrable systems obtained, we mention a deformation of the Coulomb potential with scalar potential V0 = α r + 3 2 8r 2 and spin orbital one V1 = 2r 2 .