In this paper, a lattice Boltzmann BGK (LBGK) model is proposed for simulating incompressible axisymmetric flows. Unlike other existing axisymmetric lattice Boltzmann models, the present LBGK model can eliminate the compressible effects only with the small Mach number limit. Furthermore, the source terms of the model are simple and contain no velocity gradients. Through the Chapman-Enskog expansion, the macroscopic equations for incompressible axisymmetric flows can be exactly recovered from the present LBGK model. Numerical simulations of the Hagen-Poiseuille flow, the pulsatile Womersley flow, the flow over a sphere, and the swirling flow in a closed cylindrical cavity are performed. The results agree well with the analytic solutions and the existing numerical or experimental data reported in some previous studies.