  The process $H \to J/\psi + \gamma$, where $H$ is the Higgs particle,
provides a way to probe the size and the sign of the Higgs-charm coupling. In
order to improve the theoretical control of the decay rate, we compute order
$v^4$ corrections to the decay rate based on the nonrelativistic QCD
factorization formalism. The perturbative calculation is carried out by using
automated computer codes. We also resum logarithms of the ratio of the masses
of the Higgs boson and the $J/\psi$ to all orders in the strong coupling
constant $\alpha_s$ to next-to-leading logarithmic accuracy. In our numerical
result for the decay rate, we improve the theoretical uncertainty, while our
central value is in agreement with previous studies within errors. We also
present numerical results for $H \to \Upsilon(nS) + \gamma$ for $n=1,2$, and 3,
which turn out to be extremely sensitive to the Higgs bottom coupling.
