We report nonrelativistic and scalar-relativistic coupled-cluster calculations of the copper quadrupole-coupling constants for eleven small copper-containing compounds. It is shown to be necessary to treat both electron-correlation and scalar-relativistic effects on the same footing even for a qualitatively correct description, because both effects are significant and are strongly coupled in the case of Cu electric-field gradients. We show that the three scalar-relativistic schemes employed in the present study--the leading order of direct perturbation theory, the spin-free exact two-component theory in its one-electron variant, and the spin-free Dirac-Coulomb approach--provide accurate treatments of scalar-relativistic effects for the copper compounds under study. Furthermore, we demonstrate that results close to the basis-set limit can be obtained by augmenting large uncontracted standard basis sets for copper with additional steep functions. It is also shown that high-level correlation effects (those beyond the perturbative treatment of triple excitations) make important contributions in the present case.