<p>Integrating efficient second-order (<i>χ</i>⁽²⁾) optical nonlinearity into topological integrated photonic systems presents a fundamental challenge: frequency-dependent topological bandgaps inherently impede simultaneous, robust edge states at cross-octave frequencies required for second-harmonic generation (SHG). We overcome this limitation by introducing, based on theoretical modeling and numerical simulations, dual-frequency topological bandgap engineering in a square lattice of nonlinear microresonators with synthetic magnetic fluxes. Our design achieves unidirectional edge states for both fundamental and second-harmonic frequencies. It enables efficient SHG with flux-programmable chirality, a unique consequence of <i>χ</i>⁽²⁾ nonlinearity-induced topology transitions in the system. The topological array yields an SHG efficiency enhancement of two orders of magnitude compared to a single resonator. Our design can be readily implemented by using <i>χ</i>⁽²⁾ integrated photonic platforms like thin film lithium niobate, which would unlock novel functionalities such as nonreciprocal SHG diodes, optical logic gates, and a pathway to topology-protected entangled photon sources.</p><p></p>