© 2018 authors. Published by the American Physical Society. We show that the diffractive forward production of two quarkonia, especially the χcJ and ηc states, in proton-proton or proton-ion collisions can access the Weizsäcker-Williams gluon Wigner distribution of the proton. We use the hybrid factorization approach where the collinear, double gluon PDF is applied for one of the protons and the kT-dependent (Wigner or GTMD) distribution for the other. The production of quarkonia is treated in the NRQCD framework. A particularly concise formula is obtained for double χJ=1 production.