  Derivations for the higher tensor components of the quadrupole
polarizabilities are given and their values for the metastable states of the
Ca$^+$, Sr$^+$ and Ba$^+$ alkaline earth-metal ions are estimated. We also give
the scalar quadrupole polarizabilities of the ground and metastable states of
these ions to compare our results with the previously available theoretical and
experimental results. Reasonably good agreement between our calculations with
the previous values of scalar quadrupole polarizabilities demonstrate their
correctness. The reported scalar and tensor quadrupole polarizabilities could
be very useful to estimate the uncertainties due to the gradient of the
electric fields in the clock frequencies of the above alkaline earth-metal ions
when accuracies of these frequency measurements attain below 10$^{-19}$
precision level.
