The Reststrahl spectral region of Silicon Carbide has recently attracted much attention owing to its potential for mid-infrared nanophotonic applications based on surface phonon polaritons (SPhPs). Studies of optical phonon resonances responsible for surface polariton formation, however, have so far been limited to linear optics. In this Letter, we report the first nonlinear optical investigation of the Reststrahl region of SiC, employing an infrared free-electron laser to perform second harmonic generation (SHG) spectroscopy. We observe two distinct resonance features in the SHG spectra, one attributed to resonant enhancement of the nonlinear susceptibility χ(2) and the other due to a resonance in the Fresnel transmission. Our work clearly demonstrates high sensitivity of mid-infrared SHG to phonon-driven phenomena, and opens a route to studying nonlinear effects in nanophotonic structures based on SPhPs. PACS numbers: 78.47.N-,78.30.-j,63.20.-e