An analysis is performed to investigate the effects of variable viscosity and thermal conductivity on the two-dimensional steady flow of an electrically conducting, incompressible, upper-convected Maxwell fluid in the presence of a transverse magnetic field and heat generation or absorption. The governing system of partial differential equations is transformed into a system of coupled nonlinear ordinary differential equations, and is solved numerically. Velocity and temperature fields have been computed and shown graphically for various values of the physical parameters. The local skin-friction coefficient and the local Nusselt number have been tabulated. It is found that fluid velocity decreases with an increase in the viscosity parameter and the Deborah number. It is also observed that increasing the magnetic parameter leads to a fall in the velocity and a rise in the temperature. Furthermore, it is shown that the temperature increases due to increasing the values of the thermal conductivity parameter and the heat generation parameter, while it decreases with an increase of both the absolute value of the heat absorption parameter and the Prandtl number.