After the great triumph of the Higgs discovery in 2012, the next target at the energy frontier will be to study the Higgs properties and to search for the next scale beyond the standard model of particle physics. Experimentally, the $H\to c \bar{c}$ channel would be extremely difficult to dig out because of both the weak Yukawa coupling and the daunting standard model di-jet background. We propose to test the charm-quark Yukawa coupling at the LHC and future hadron colliders with the Higgs boson decay to $J/\psi$ via the charm-quark fragmentation. Using the non-relativistic quantum chromodynamics, we study the charmonia production via the Higgs boson decay channel $ H \to c \ \bar{c} + J/\psi $(or $ \eta_c $), where both the color-singlet and color-octet contributions are considered. Our result opens another door to improve determinations at the LHC of the Higgs Yukawa couplings: the final state from this decay mode is quite distinctive with $J/\psi\to e^+e^-,\, \mu^+\mu^-$ and the branching fraction is enhanced by the charm-quark fragmentation mechanism.