Physics beyond the standard model (SM) can be parametrized with an effective Lagrangian that respects the symmetries of the standard model and contains many operators of dimension six. We consider the subset of these operators that is responsible for flavor diagonal anomalous color magnetic and electric dipole couplings between quarks and gluons. Invariance of these operators under the SM implies that they contribute to Higgs boson production at the LHC, and we study the corresponding constraints that can be placed on them. For the case of the top quark we first review constraints from top-quark pair production and decay, and then compare them to what can be achieved by studying tt̄h production. We also constrain the corresponding couplings for b quarks and light quarks by studying pp→bb̄h and pp→hX, respectively.