An ab initio coupled cluster singles and doubles (CCSD) investigation of the electric-field- gradient-induced birefringence (EFGB) of a polar molecule, CO, is presented. The so-called "effective quadrupole center" (EQC), the origin to which the quadrupole moment deduced from EFGB experiments refers for dipolar molecules and which experimentalists cannot determine directly, is computed within two different semiclassical theories of the EFGB. The temperature independent frequency dependent contribution to the birefringence is determined as well. The molecular electric quadrupole moment with the origin at the EQC is determined within the two theories via an accurate value of the quadrupole moment referring to the center of mass, obtained using well-established hierarchies of basis sets and wave function models, and the calculated EQCs. The final theoretical values result as the basis set limit at the CCSD(T) level completed with estimates for relativistic effects, correlation contributions beyond CCSD(T), rovibrational effects, and residual errors in the EQC. The comparison of the theoretical results with the corresponding experimentally deduced values sheds some light on the differences between the two semiclassical approaches to the EFGB.