One of candidates for the generation mechanism of high linearly polarized γ rays in γ-ray bursts is synchrotron radiations from high energy electrons under strong magnetic fields. If this scenario is true, Hermite Gaussian (HG) wave photons, which are one of high-order Gaussian modes, are also generated by high-order harmonic radiations in strong magnetic fields. The HG wave γ rays propagating along the z-direction have quantum numbers of nodes of nx and ny in the x- and y-directions, respectively. We calculate the differential cross sections for Compton scattering of photons described by HG wave function in the framework of relativistic quantum mechanics. The results indicate that it is possible to identify the HG wave photon and its quantum numbers nx and ny and by measuring the azimuthal angle dependence of differential cross section or the energy spectra of the scattered photon as a function of the azimuthal angle.