Forward photoproduction of J / ψ can be used to extract generalized parton distributions (GPDs) of gluons. We analyze the process at twist-3 level and study relevant classifications of twist-3 gluon GPDs. At leading power or twist-2 level the produced J / ψ is transversely polarized. We find that at twist-3 the produced J / ψ is longitudinally polarized. Our study shows that in the high-energy limit the twist-3 amplitude is only suppressed by the inverse power of the heavy quark mass relatively to the twist-2 amplitude. This indicates that the power correction to the cross-section of unpolarized J / ψ can have a sizable effect.The poles in the hard factor convoluted with twist-3 gluon GPDs may concern the GPDs discontinuities, which will be briefly discussed in this work. We have also derived the amplitude of the production of h c at twist-3, but the result contains end-point singularities. The production of other quarkonia has been briefly discussed.