Control over dispersion of nanoparticles in polymer solutions through porous media is important for subsurface applications such as soil remediation and enhanced oil recovery. Dispersion is affected by the spatial heterogeneity of porous media, the non-Newtonian behavior of polymer solutions, and the Brownian motion of nanoparticles. Here, we use the Euler-Lagrangian method to simulate the flow of nanoparticles and inelastic non-Newtonian fluids (described by Meter model) in a range of porous media samples and injection rates. In one case, we use a fine mesh of more than 3 million mesh points to model nanoparticles transport in a sandstone sample. The results show that the velocity distribution of nanoparticles in the porous medium is non-Gaussian, which leads to the non-Fickian behavior of nanoparticles dispersion. Due to pore-space confinement, the long-time mean-square displacement of nanoparticles depends nonlinearly on time. Additionally, the gradient of shear stress in the pore space of the porous medium dictates the transport behavior of nanoparticles in the porous medium. Furthermore, the Brownian motion of nanoparticles increases the dispersion of nanoparticles along the longitudinal and transverse direction.