Summary: Many researchers have expanded a new form of fractional diffusion models named the temporal-space fractional Bloch-Torrey equation (TSF-BTE) to evaluate the diffusion structure of human brain tissues as well as prepare additional insight into other studies of cells and tissues and the microenvironment. The main objective of this paper is to present an effective computational method of solving such models in two dimensions. The temporal and spatial directions are based on the Caputo and the Riemann-Liouville fractional derivative, respectively. The presented numerical scheme is derived from the following manners: at first, the semi-discrete is constructed in the temporal based on a quadratic interpolation with accuracy order \(\mathcal{O}(\tau^{2-\alpha})\) and secondly, the unconditional stability and convergence order are analyzed. For the constructed full-discrete scheme, the spatial derivative terms approximated with the helping of the collocation method based on the Legendre basis. Finally, to illustrate the high precision of the proposed design, we use some test problems. Furthermore, the obtained results are compared with some other techniques under which the suggested methodology is highly accurate and feasible.