What is the magnetic quadrupole operator? To find magnetic or electrical moments in quantum theory we must calculate the expectation value of an appropriate operator. The dipole operators are similar and are easy to find but I couldn't find the form for magnetic quadrupole moment.
 A: The magnetic quadrupole moment tensor is given by
$$m_{ij}=\left\langle \frac{2}{3}\left(\mathbf{r}\times\mathbf{J}\right)_i r_j \right\rangle,$$
in analogy with the magnetic dipole moment vector
$$m_i=\left\langle \frac{1}{2}\left(\mathbf{r}\times\mathbf{J}\right)_i  \right\rangle.$$
The magnetic field at a point $\mathbf{R}$ is then, up to quadrupole order,
$$
B_i(\mathbf{R})=\frac{\mu_0}{4\pi}\left[ m_j
\frac{R_iR_j-3R^2\delta_{ij}}{R^5}
+\frac{3m_{jk}}{2R^7}\left(5R_iR_jR_k -R^2(R_i\delta_{jk}+R_j\delta_{ki}+R_k\delta_{ij}) \right)
\right].
$$
This link has what looks like a good exposition of the subject (including the formulae above and a pretty comprehensive list of references), chapter 1 of

R.E. Raab and O.L. de Lange. Multipole Theory In Electromagnetism: Classical, Quantum, And Symmetry Aspects, With Applications (International Series of Monographs on Physics), 1st Edition (Clarendon Press - Oxford University Press, Oxford, 2005).

There's a few red flags here, though, which I can't quite see through. I am unsure whether a symmetric + traceless magnetic quadrupole moment is possible, and the expression above definitely has commutation issues in the quantum case, which you should be careful with. On the other hand, the electron spin fortunately doesn't contribute to anything above dipole order, for symmetry reasons, so you're dealing only with the position coordinates and orbital angular momentum where needed.
The current $\mathbf{J}$ is proportional to the particle's velocity and this may or may not be proportional to the canonical momentum (in which case, for example, $\mathbf{m}\propto\mathbf{L}$), depending on what gauge you're working in. Be careful!
Finally, magnetic quadrupoles are really close to the edge of our measurement capabilities, and certainly nothing above the E3/M2 terms has been observed experimentally in atomic experiments as far as I'm aware (but see Ben Crowell's comment for nuclear physics). Electric octopole transitions are very strongly forbidden, with lifetimes of seconds or even years (!), but they have been used to great effect in precision metrology experiments on trapped Ytterbium ions (see e.g. this answer). In general, though, unambiguous detection of effects at this order requires transitions which are forbidden at both the E1 and the E2/M1 orders, and those usually turn out to be hard to address.
