In this work, we tackle the problem of hyperspectral unmixing by departing from the usual linear model and focusing on a linear-quadratic (LQ) one. The algorithm we propose, coined Successive Nonnegative Projection Algorithm for Linear Quadratic mixtures (SNPALQ), extends the Successive Nonnegative Projection Algorithm (SNPA), specifically designed to address the unmixing problem under a linear non-negative model and the pure-pixel assumption (a.k.a. near-separable assumption). By explicitly modeling the product terms inherent to the LQ model along the iterations of the SNPA scheme, the nonlinear contributions of the mixing are mitigated, thus improving the separation quality. The approach is shown to be relevant in realistic numerical experiments, which further highlight that SNPALQ is robust to noise.