<p>The J-matrix method is an algebraic method that was developed long ago to treat quantum scattering for short range linear potentials. Here and for the first time in the literature, we introduce a nonlinear extension of the method by considering a quadratic self-interaction model. The formulation relies predominantly on the linearization of products of orthogonal polynomials. For simplicity, we present a toy model as an illustrative example and obtain the nonlinear scattering matrix.</p>