We calculate (in the context of the standard model) the rate for radiative decay of a heavy-quark–antiquark 3S1 bound state into Higgs-boson pairs, i.e., Γ(V→HHγ). This process has contributions from diagrams containing the three-Higgs-boson self-coupling. We investigate whether that coupling is measurable in decays of the t-quarkonium system or a hypothetical, heavy, fourth-generation b’-quarkonium. We also briefly discuss the prospects for measuring such trilinear Higgs-boson couplings in heavy-quarkonium decays for two-Higgs-doublet theories.