The time fractional Cattaneo-Christov flux heat model is first introduced to investigate the flow and heat transfer of Maxwell viscoelastic fluid past a vertical flat plate. Fractional constitutive relation and Cattaneo-Christov heat flux model are applied to construct the governing boundary layer equations of momentum and energy, which are nondimensionalized by new dimensionless variables and solved numerically. The results indicate that there exist intersections on velocity and temperature profiles for different values of Prandtl number when the fractional Cattaneo-Christov flux heat model is considered.