Mode-locked vertical external-cavity semiconductor lasers (VECSELs) are a wavelength-versatile laser that relies on a semiconductor saturable absorber mirror (SESAM) to initiate pulsed emission while simultaneously significantly influencing the pulse's properties. A SESAM can be characterized using a nonlinear reflectivity setup, realized here in the red spectral range around 660 nm and achieving a moderate peak-to-peak variation of 0.17%. We use our home-built mode-locked VECSEL to reach a high maximum fluence up to 430 µJ/cm2 with strongly chirped 7.5 ps pulses. This allows the first-of-its-kind characterization of GaInP quantum well SESAMs, thereby revealing saturation fluences of 38 µJ/cm2 and 34 µJ/cm2 with modulation depths of 5% and 10.3% for SESAMs comprising one or two active quantum wells, respectively. For all structures, a nonsaturable loss of 2.8% is found and attributed to scattering loss.