Dielectric gratings that couple optical fibers and planar waveguide circuits are key for optical-to-electronic (electronic-to-optical) signal conversion, but their applicability to platforms that require broader bandwidths and higher capacity is limited by their single-wavelength response. Herein, we present the design of a quasi-periodic grating coupler with multiband fiber-to-waveguide (waveguide-to-fiber) coupling response, where the grating consists of a periodic repetition of unit cells made of alternating silicon and air grooves according to the Fibonacci sequence. Through finite-difference time-domain (FDTD) calculations, we show that this new device could be used for coupling multi-wavelength fiber modes in a single grating structure. The results were obtained for fibers operating in the wavelength range from 1000 nm to 2000 nm, but the concept can be readily extended to other frequency ranges. Moreover, the allowed modes in the grating are almost insensitive to fiber misalignments and small fabrication errors for high Fibonacci steps, which is useful when alignment of optical components is impractical. It is hoped that properly designed gratings overlapping multiple modes may lead to ultra-broadband fiber-waveguide couplers that can cope with the growing demand for higher capacity and bandwidth in optical communications.