The study of the boundary layer flow is considered as one of the fundamental issues in fluid mechanics that attracted many of the researchers’ attention in this field. The most of previous researches on boundary layer problem are limited to Newtonian fluids and a few numbers of researches are considered the non-Newtonian fluids. The main objective of this research is to better recognize of the viscoelastic properties effect on characteristics of the boundary layer. In this study, the hydrodynamic and thermal boundary layer of viscoelastic Falkner-Skan problem is investigated numerically. Here, the second order model has been used as the viscoelastic constitutive equation and analyzed with MATLAB software. Both constant temperature and constant heat flux at walls are used as thermal boundary conditions. The effect of Reynolds number, the coefficient of the first normal stress difference, Prandtl number and wedge shape factor on the thickness of dynamic and heat transfer boundary layer, momentum thickness, displacement thickness, drag coefficient and Nusselt number are studied. The effects of both adverse pressure gradient and favorable pressure gradient in heat and hydrodynamics boundary layer characteristics are investigated. The magnitude of the non-Newtonian hydrodynamic and heat boundary layer are found to increase with increasing the coefficient of the first normal stress and finally at constant pressure gradient, average Nusselt number decreases along plate by increasing first normal stress coefficient.