We analyze the temporal dynamics of an optically-pumped quantum well vertical external-cavity surface-emitting laser (VECSEL) with a Semiconductor Saturable Absorber Mirror (SESAM) using the time series obtained when varying the pump power. We unveil the quasiperiodic route to chaos in the system by characterizing the Fourier spectra, the attractors in phase space, and the Lyapunov exponents for each temporal behavior observed: periodicity, quasiperiodicity, and chaos. Thus, we provide a complete description of this experimental observation of the route to chaos in a VECSEL-SESAM system.