  We evaluate the electric quadrupole moments of the ${^1}\!S_0$ and
${^3}\!P_0$ clock states of $^{27}$Al$^+$ and $^{115}$In$^+$. To capture all
dominant contributions, our analysis extends through third order of
perturbation theory and includes hyperfine coupling of the electrons to both
the magnetic dipole and electric quadrupole moments of the nucleus. For
$^{27}$Al$^+$, a fortuitous cancellation leads to a suppressed frequency shift.
This should allow for continued improvement of the clock without special
techniques to control or cancel the shift, such as the averaging schemes that
are critical to other optical ion clocks.
