In many integrated optics systems, grating couplers are a key component of interfacing the external light source with in-plane photonic devices. Grating couplers with dual-band capability are often desired for expanding the operation spectrum of photonic systems. Here, we propose and theoretically investigate, for the first time, a 4.95 µm-8.5 µm dual-band grating coupler on a Ge-on-SOI platform. In addition to conventional structures, Bragg gratings are introduced to two wavelength division directions for crosstalk suppression. With this design, the simulated coupling efficiencies have respectively reached 59.93% and 46.38% for the 4.95 µm and 8.5 µm bands. This mid-infrared dual-band grating coupler may be useful for defense and environmental monitoring applications.