The low-energy capture of homonuclear diatoms by ions is due mainly to the long-range part of the interpartner potential with leading terms that correspond to charge-quadrupole interaction and charge-induced dipole interaction. The capture dynamics is described by the perturbed-rotor adiabatic potentials and the Coriolis interaction between manifold of states that belong to a given value of the intrinsic angular momentum. When the latter is large enough, it can noticeably affect the capture cross section calculated in the adiabatic channel approximation due to the gyroscopic property of a rotating diatom. This paper presents the low-energy (low-temperature) state-selected partial and mean capture cross sections (rate coefficients) for the charge-quadrupole interaction that include the gyroscopic effect (decoupling of intrinsic angular momentum from the collision axis), quantum correction for the diatom rotation, and the correction for the charge-induced dipole interaction. These results complement recent studies on the gyroscopic effect in the quantum regime of diatom-ion capture