In this article, a new algorithm is proposed to solve the nonlinear fractional-order one-dimensional solute transport system. The spectral collocation technique is considered with the Fibonacci polynomial as a basis function for the approximation. The Fibonacci polynomial is used to obtain derivative in terms of an operational matrix. The proposed algorithm is actually based on the fact that the terms of the considered problem are approximated through a series expansion of double Fibonacci polynomials and then collocated those on specific points, which provide a system of nonlinear algebraic equations which are solved by using Newton's method. To validate the precision of the proposed method, it is applied to solve three different problems having analytical solutions. The comparison of the results through error analysis is depicted through tables which clearly show the higher accuracy of order of convergence of the proposed method in less central processing unit (CPU) time. The salient feature of the article is the graphical exhibition of the movement of solute concentration for different particular cases due to the presence and absence of reaction term when the proposed scheme is applied to the considered nonlinear fractional-order space–time advection–reaction–diffusion model.