The scattering cross sections of (anti)neutrinos off the detector material in liquid xenon detectors is of great interest in both dark matter searches and neutrino physics. Here it is necessary to use calculated cross sections because directly observed experimental ones do not exist. In this work we provide the scattering cross sections for charged-current (anti)neutrino scattering off the even–odd xenon nuclei ${}^{129,131}$Xe. To do this we use G-matrix-based two-nucleon interactions in large single-particle valence spaces. We model the nuclei involved using the Microscopic Quasiparticle-Phonon Model, constructing the states of the even–odd and odd–even nuclei from BCS (Bardeen–Cooper–Schrieffer) quasiparticles and QRPA (Quasiparticle Random Phase Approximation) phonons of the neighbouring even–even nuclei. Such calculations have been done before for ${}^{129,131}$Xe, but here we go beyond earlier works by including the effect of the meson-exchange two-body currents (2BC) that can naturally explain a part of the quenching of the weak axial-vector coupling $g_A$. A comparison to a phenomenologically quenched one-body-currents-only (1BC-only) model shows that the inclusion of the 2BC can account for a majority of the quenching of $g_A$.