  We investigate the SM Higgs decays into heavy quarkonia J/\psi and \Upsilon
in both color-singlet and color-octet mechanisms. It is found that in J/\psi
production the contribution through color-octet processes overwhelm that in
color-singlet processes all the intermediate region M_Z<M_{H^0}<2 M_w and the
fraction ratio is comparatively large, but in \Upsilon production the
contribution through color-octet mechanism is negligible.
