In this paper, accurate and efficient simulation of cell motion in a biological fluid flow is investigated. The membrane of a moving cell is represented by a thin shell composed of incompressible neo-Hookean elastic materials and the liquids around the membrane are approximated as incompressible Newtonian flows with low Reynolds numbers. The biofluid mechanics is approximated by the Stokes flow equations. A low-order BEM model is developed for the two biological fluids coupled at the membrane surface. The moving boundary problem in fluid mechanics can be effectively solved using the BEM with a GMRES solver. The FEM model based on a flat thin shell element is further developed to predict the membrane load due to the large deformation of a moving cell. Computational efficiency is greatly improved due to the one-dimensional reduction in the present BEM and FEM models. The BEM solver for the biological fluids is coupled with the FEM solver for the cell membrane at the membrane surface. The position of the membrane surface nodes is advanced in time by using the classical fourth-order Runge-Kutta method. Numerical instability is avoided by using a relatively small time step. Further numerical instabilities in the FEM solver is alleviated by using various techniques. The present method is applied to the FSI problems of cell motion in a cylindrical flow. Numerical examples can illustrate the distinct accuracy, efficiency and robustness of the present method. Furthermore, the importance of bending stiffness of a cell membrane for stable cell motion simulation is emphasized. It is suggested that the present approach be an appealing alternative for simulating the fluid-structure interaction of moving cells.