The author considers the Mathieu differential equation
where is a a given constant while is the (known) spectral parameter, and gives some explicit expressions for the Fourier coefficients of its quasi-periodic solutions corresponding to the characteristic multiplier with . The main results are stated in two theorems which concern the cases and , respectively. The results involve certain sets of polynomials which have connections with the Klein-Gordon equation. Notice that the Fourier coefficients in question satisfy a three-term recurrence relation whose explicit solution is not known in terms of known functions.