  In the context of perovskite quantum paraelectrics, we study the effects of a
quadrupolar interaction $J_q$, in addition to the standard dipolar one $J_d$.
We concentrate here on the nonlinear dielectric response $\chi_{P}^{(3)}$, as
the main response function sensitive to quadrupolar (in our case
antiquadrupolar) interactions. We employ a 3D quantum four-state lattice model
and mean-field theory. The results show that inclusion of quadrupolar coupling
of moderate strength ($J_q \sim {{1}\over{4}} J_d$) is clearly accompanied by a
double change of sign of $\chi_{P}^{(3)}$ from negative to positive, near the
quantum temperature $T_Q$ where the quantum paraelectric behaviour sets in. We
fit our $\chi_{P}^{(3)}$ to recent experimental data for SrTiO$_3$, where the
sign change is identified close to $T_Q \sim 37 K$.
