Stimulated Brillouin scattering (SBS) in a perfluorinated graded-index polymer optical fiber (POF) with 120 μm core diameter was experimentally observed for the first time, to the best of our knowledge, at 1.55 μm wavelength with the pump-probe technique. Compared to spontaneous Brillouin scattering previously reported, the Brillouin gain spectrum (BGS) was detected with an extremely high signal-to-noise ratio, even with a short POF (1 m) and scrambled polarization state. We also investigated the BGS dependences on probe power and temperature, which indicate that SBS in a POF measured with this technique can be utilized to develop high-accuracy temperature sensing systems.