Photonic quasicrystals are poised to transform the field of nonlinear light-matter interactions due to their ability to support an unlimited number of combinations of wavevectors in their reciprocal lattices. Such greatly enhanced flexibility enabled by k-space engineering makes photonic quasicrystals a promising platform for novel approaches to multi-wavelength conversion, supercontinuum generation, and development of classical and quantum optical sources. Here, we develop a new design method for nonlinear photonic quasicrystals, consisting of a combination of one nonlinear material and one linear material that can simultaneously fulfill phase-matching conditions for a desired number of nonlinear optical interactions as long as the frequencies of the interacting waves are outside of the bandgaps of the quasicrystal structure. Our approach provides enhanced design flexibility, enabling new pathways to designing compact, integrated nonlinear photonic devices and systems on a chip.