Magnetic skyrmions have attracted great research interest in recent years due to their exotic physical properties, scientific merit and potential applications in modern technology. Here, we apply a quantum computational method to investigate the spin configurations of FeCoSi-like quasi-two-dimensional ferromagnetic system with co-existence of Dzyaloshinskii–Moriya and Heisenberg exchange interactions. We find that within a weak magnetic field perpendicular to the film plane, skyrmion crystal (SkX) of hexagonal-close-packed pattern can be induced, the spin configurations evolve with applied magnetic field and temperature. This quantum model, if scaled, is able to qualitatively reproduce the experimental results of SkX with long periodicity. Especially, when the skyrmion size is around a few nano-meters in diameter, or more general, when the characteristic length of the material approaches the lattice scale, the quantum model is expected to be more accurate than the classical ones.