The need for a consistent and reliable energy supply to improve productivity is steadily increasing both in industrial and residential settings. The fulfillment of this requirement can be effectively achieved by implementing quadratic thermal radiation, such as utilizing an appropriate thermoelectric generator to supply power and optimizing the optical heat process. This research investigates the three different thermal radiation models, namely linear, quadratic, and nonlinear thermal radiation on the two-phase boundary layer flow of a dusty nanofluid across a vertically moving flat plate. The nanofluid is composed of aluminum oxide (Al203) and nanoparticles suspended in water (H20) based fluid. Furthermore, quadratic Boussinesq approximation and Prandtl's boundary layer approximation are utilized. The Chebyshev collocation spectral method is employed to address the non-linear problem. The effect of different parameters, such as thermal Grashof number, momentum dust, magnetic field parameter on the fluid flow, and temperature profiles are considered to visualize the findings. The findings show that linear thermal radiation exhibits the lowest thermal transport, followed by quadratic thermal radiation, with nonlinear thermal radiation demonstrating the highest thermal transport. The velocity of the dusty nanoliquid is decreased when a transverse magnetic field is present. Additionally, it is observed that the existence of aluminum oxide nanoparticles of 3% volume concentration in particulate H20 effectively increases the thermal transfer of the fluid system.