In the paper, we present a calculation of the decay widths for the Higgs boson decays to the Bc, B * c , Bc(2 1 S0) and B * c (2 3 S1) mesons using the fragmentation-function approach. In the calculation, the fragmentation functions up to order α 3 s based on the nonrelativistic QCD factorization theory are used, and the decay widths for H → Q + X and H → g + X at the partonic level are calculated up to order αs. The large logarithms of m 2 H /m 2 Bc are resummed up to next-to-leading logarithmic accuracy by solving the evolution equations for the running quark masses and the fragmentation functions. Compared to the leading-order decay widths based on the nonrelativistic QCD approach, the decay widths based on the fragmentation-function approach that include the higher-order QCD corrections are reduced significantly. Our numerical results show that there are about 1.2 × 10 5 Bc events via the Higgs decays to be produced at the HL-LHC with 3ab −1 , and about 1.6 × 10 6 Bc events via the Higgs decays to be produced at the HE-LHC with 15ab −1 .