  We derive a quantum-mechanical formula of the orbital magnetic quadrupole
moment (MQM) in periodic systems by using the gauge-covariant gradient
expansion. This formula is valid for insulators and metals at zero and finite
temperature. We also prove a direct relation between the MQM and
magnetoelectric (ME) susceptibility for insulators at zero temperature. It
indicates that the MQM is a microscopic origin of the ME effect. Using the
formula, we quantitatively estimate these quantities for room-temperature
antiferromagnetic semiconductors BaMn$_2$As$_2$ and CeMn$_2$Ge$_{2 - x}$Si$_x$.
We find that the orbital contribution to the ME susceptibility is comparable
with or even dominant over the spin contribution.
