  A supersymmetric extension of the standard model based on the discrete $Q_6$
family symmetry is considered, and we investigate flavor-changing neutral
current (FCNC) processes, especially those mediated by heavy flavor-changing
neutral Higgs bosons. Because of the family symmetry the number of the
independent Yukawa couplings is smaller than that of the observed quantities
such as the Cabibbo-Kobayashi-Maskawa matrix and the quark masses, so that the
FCNCs can be parametrized only by the mixing angles and masses of the Higgs
fields. We focus our attention on the mass differences of the neutral $K, D$
and $B$ mesons. All the constraints including that from the ratio $\Delta
M_{B_s}/\Delta M_{B_d}$ can be satisfied, if the heavy Higgs bosons are heavier
than $\sim 1.5$ TeV. If the constraint from $\Delta M_K$ is slightly relaxed,
the heavy Higgs bosons can be as light as $\sim 0.4$ TeV, which is within the
accessible range of LHC.
