<p>The steady, two-dimensional magnetohydrodynamic (MHD) boundary layer flow and heat transfer of a piezoviscous fluid over a permeable stretching or contracting wedge are investigated in this study. Unlike classical Newtonian fluids, the dynamic viscosity of the working fluid is assumed to be exponentially dependent on the local mechanical pressure; consequently, the fluid’s rheological response is fundamentally altered, and the deviatoric stress is closely coupled to the pressure field. A transverse magnetic field is incorporated, and variable thermal conductivity is accounted for in the analysis, thereby enhancing the physical fidelity of the heat transfer model. The complex, highly coupled governing partial differential equations—representing the conservation of mass, momentum, and thermal energy—are simplified using Prandtl’s boundary layer approximations. Due to the explicit streamwise dependence of the piezoviscous viscosity, a global similarity solution cannot be achieved. Instead, appropriate local similarity transformations are employed to reduce the partial differential equations into a system of coupled, nonlinear ordinary differential equations. These equations, generalized forms of the classical Falkner–Skan problem, are solved numerically using a robust Runge–Kutta integration scheme. The effects of the pressure–viscosity coefficient, magnetic interaction parameter, wedge angle, and variable thermal conductivity on the velocity and temperature profiles, as well as on key engineering quantities like the skin friction coefficient and local Nusselt number, are systematically analyzed. Critical insights into the thermo-fluidic behavior of fluids under extreme pressure environments, where the assumption of constant viscosity is physically restrictive, are provided by these findings.</p>