Linear aero-thermo-elastic stability of functionally graded material (FGM) plates subjected to aerothermal heating is investigated with emphasis on heat-flux-driven thermal gradients. The through-thickness temperature field is obtained by solving the one-dimensional steady-state heat conduction equation under prescribed heat-flux and temperature boundary conditions, representing realistic aerodynamic heating environments. The resulting nonlinear temperature distribution is incorporated into the plate constitutive relations to account for thermally induced in-plane stresses. Linear thermal buckling and flutter characteristics are analyzed within an eigenvalue framework. The structural behavior of the FGM plate is modeled using the first-order shear deformation plate theory, while the unsteady aerodynamic pressure associated with supersonic flow is evaluated using piston theory. Critical thermal buckling boundaries are determined in terms of heat-flux parameters, and dynamic instability limits are identified through complex eigenvalue analysis. The influence of material gradation, geometric parameters, and aerodynamic loading on the aero-thermo-elastic stability boundaries is examined. The results demonstrate that heat-flux-driven thermal gradients significantly affect both static and dynamic stability characteristics, leading to reduced critical aerodynamic pressure and thermal buckling thresholds compared to idealized prescribed-temperature models. The present study provides a more physically realistic assessment of the aero-thermo-elastic stability of FGM plates under aerothermal heating.