  The magnetoelectric effect has been actively studied in multiferroics since
the first observation in an antiferromagnetic, $\mathrm{Cr_2O_3}$. This effect
appears in systems without spatial inversion symmetry and time-reversal
symmetry and is sensitive to detecting magnetic quadrupole moments. It is often
discussed as inducing spin magnetizations; however, the orbital magnetoelectric
effect in metals has recently attracted much attention since its observation in
$\mathrm{MoS_2}$ and twisted bilayer graphene. In this work, we propose the
full quantum formalism for the temperature gradient induced-orbital
magnetoelectric effect (orbital gravito-ME effect). The effect consists of two
parts, i.e., an extrinsic part and an intrinsic part. We demonstrate that the
intrinsic part needs a correction from the orbital magnetic quadrupole moment
besides the usual Kubo formula to avoid an unphysical divergence at zero
temperature and to satisfy the Mott relation. Furthermore, we show the
classification table with the magnetic point group for the intrinsic and
extrinsic effects. Finally, we analyze the intrinsic part in a
$\mathcal{PT}$-symmetric model exhibiting an orbital magnetization order, i.e.,
a loop current order, and demonstrate the enhancement near Dirac points. We
believe that these results will contribute to the detection and usage of
orbital magnetic moments beyond spin moments.
