In this Letter, we present an analytical and numerical investigation to characterize the formation of quadratic doubly periodic waves originating from coherent modulation instability in a dispersive quadratic medium in the regime of cascading second-harmonic generation. To the best of our knowledge, such an endeavor has not been undertaken before, despite the growing relevance of doubly periodic solutions as the precursor of highly localized wave structures. Unlike the case with cubic nonlinearity, the periodicity of quadratic nonlinear waves can also be controlled by the wave-vector mismatch in addition to the initial input condition. Our results may impact widely on the formation, excitation, and control of extreme rogue waves and the description of modulation instability in a quadratic optical medium.