The production of the heavy quarkonium, i.e., jðcbÞ½ni (or jðbcÞ½ni), jðccÞ½ni, and jðbbÞ½ni-quarkonium [jðQQ 0 Þ½ni-quarkonium for short], through Higgs H 0 boson semiexclusive decays is evaluated within the nonrelativistic quantum chromodynamics (NRQCD) framework, where [n] stands for the production of the two color-singlet S-wave states, jðQQ 0 Þ½ 1 S 0 1 i and jðQQ 0 Þ½ 3 S 1 1 i, the production of the four color-singlet P-wave states, i.e., jðQQ 0 Þ½. Moreover, according to the velocity scaling rule of the NRQCD, the production of the two color-octet components, jðQQ 0 Þg½ 1 S 0 8 i and jðQQ 0 Þg½ 3 S 1 8 i, are also taken into account. The "improved trace technology" to derive the simplified analytic expressions at the amplitude level is adopted, which shall be useful for dealing with these decay channels. If all higher heavy quarkonium states decay completely to the ground states, it should be obtained ΓðH