The influence of oscillating quadrupole fields on atomic energy levels is examined theoretically and general expressions for the quadrupole matrix elements are given. The results are relevant to any ion-based clock in which one of the clock states supports a quadrupole moment. Clock shifts are estimated for $^{176}$Lu$^+$ and indicate that coupling to the quadrupole field would not be a limitation to clock accuracy at the $lesssim10^{-19}$ level. Nevertheless, a method is suggested that would allow this shift to be calibrated. This method utilises a resonant quadrupole coupling that enables the quadrupole moment of the atom to be measured. A proof-of-principle demonstration is given using $^{138}$Ba$^+$, in which the quadrupole moment of the $D_{5/2}$ state is estimated to be $Theta=3.229(89) e a_0^2$.
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