Q-calculus plays an extremely important role in mathematics and physics, especially in quantum physics, spectral analysis and dynamical systems. In recent years, many scholars are committed to the research of nonlinear quantum difference equations. However, there are few works about the nonlinear $q-$difference equations with $p$-Laplacian. In this paper, we investigate the solvability for nonlinear second-order quantum difference equation boundary value problem with one-dimensional $p$-Laplacian via the Leray-Schauder nonlinear alternative and some standard fixed point theorems. The obtained theorems are well illustrated with the aid of two examples.