The effects of quadratic thermal radiation and quadratic Boussinesq approximation are investigated on the heat transport of a 36 nm Al2O3 − H2O nanofluid over a vertical plate. The modified Buongiorno model is used in the analysis that includes the effectual thermophysical properties of the nanofluid and the key slip mechanisms. Experimentally verified correlations are used for the thermophysical properties. The reduced nonlinear differential problem is solved numerically by the Finite Difference Method (FDM). Flow profiles are displayed and analyzed for changes in dimensionless parameters. Further, the heat transfer flux at the wall is analyzed for interactive impacts of the buoyancy ratio, Brownian random motion, and thermophoresis parameters using the face-centered Central Composite Design (CCD) of the Response Surface Methodology (RSM). A sensitivity analysis is carried out for the heat transfer flux of the nanoliquid. Quadratic thermal radiation was found to improve the temperature profile. Furthermore, the mechanisms of Brownian random motion and thermophoresis have a negative sensitivity towards the rate of heat transfer. In various thermal applications like solar collectors, the density variation in terms of temperature differences is significantly high. Such phenomena can be accurately modeled by utilizing the quadratic Boussinesq approximation and the novel quadratic thermal radiation aspect.