The quadrupole operator is a special case of the crystal-field operator. The operator is defined as: \begin{equation} Q_{ij} = 3r_ir_j/r^2 \hat{r}\cdot\hat{r}\delta_{i,j}. \end{equation} Whereby we need to note that we here only take the angular part of a quadrupole. It's “strength”, i.e. the expectation value of $r^2$ is omitted from our definition.