Different kinds of wave packet transforms are widely used for extracting multi-scale structures in signal processing tasks. This paper introduces the quantum circuit implementation of a broad class of wave packets, including Gabor atoms and wavelets, with compact frequency support. Our approach operates in the frequency space, involving reallocation and reshuffling of signals tailored for manipulation on quantum computers. The resulting implementation is different from the existing quantum algorithms for spatially compactly supported wavelets and can be readily extended to quantum transforms of other wave packets with compact frequency support.