Let p and q be polynomials with degree 2 over an arbitrary field F, such that p(0)q(0)≠0. A square matrix with entries in F is called a (p,q)-product when it can be split into AB for some pair (A,B) of square matrices such that p(A)=0 and q(B)=0.A (p,q)-product is called regular when none of its eigenvalues is the product of a root of p and of a root of q. A (p,q)-product is called exceptional when all its eigenvalues are products of a root of p and of a root of q. In a previous work [6], we have shown that the study of (p,q)-products can be entirely reduced to the one of regular (p,q)-products and to the one of exceptional (p,q)-products. Moreover, regular (p,q)-products have been characterized in [6] thanks to structural theorems on quaternion algebras, giving the problem a completely unified treatment.The present article completes the study of (p,q)-products by obtaining a complete characterization of exceptional (p,q)-products. Beforehand, only very special cases in this problem had been solved, most notably the ones of products of two involutions, products of two unipotent elements of index 2, and products of a unipotent element of index 2 with an involution.The study involves tools and strategy that are similar to the ones used for the exceptional (p,q)-sums undertaken in a recent article [7].