We use the first integral method to construct closed-from exact solutions of the density dependent Fitzhugh-Nagumo telegraph equation. While this method is very useful for obtaining formal solutions, many of the solutions are not physically relevant. Hence, we carefully determine which of these solutions is physically useful for the problem. For standard diffusion and linear density dependent diffusion, we obtain physically meaningful exact solutions. Interestingly, the case with linear density dependence shows strong qualitative agreement the standard diffusion case. For both cases, the wave solutions result in an initial concentration which either dissipates or tends to some finite positive value for large time. While one can obtain wave solutions with general wave speeds for the standard diffusion case, when there is density dependence in diffusion, the resulting wave speeds will take specific values depending on the values of the model parameters. For stronger density dependent diffusion involving quadratic and higher order functions of the concentration, we show that the equation is not integrable, and exact solutions should not be expected for such cases.