We analyse the Fundamental Plane (FP) relation of 39993 early-type galaxies (ETGs) in the optical (griz) and 5080 ETGs in the near-infrared (NIR; YJHK) wavebands, forming an optical+NIR sample of 4589 galaxies. We focus on the analysis of the FP as a function of the environment where galaxies reside. We characterize the environment using the largest group catalogue, based on 3D data, generated from the Sloan Digital Sky Survey at low redshift (z < 0.1). We find that the intercept `c' of the FP decreases smoothly from high- to low-density regions, implying that galaxies at low density have on average lower mass-to-light ratios than their high-density counterparts. The `c' also decreases as a function of the mean characteristic mass of the parent galaxy group. However, this trend is weak and completely accounted for by the variation of `c' with local density. The variation of the FP offset is the same in all wavebands, implying that ETGs at low density have younger luminosity-weighted ages than cluster galaxies, consistent with the expectations of semi-analytical models of galaxy formation. We measure an age variation of ~0.048 dex (~11 per cent) per decade of local galaxy density. This implies an age difference of about 32 per cent (~3Gyr) between galaxies in the regions of highest density and the field. We find the metallicity decreasing, at ~2σ, from low to high density. We also find 2.5σ evidence that the variation in age per decade of local density augments, up to a factor of 2, for galaxies residing in massive relative to poor groups. The velocity dispersion slope of the FP, a, tends to decrease with local galaxy density, with galaxies in groups having smaller a than those in the field, independent of the waveband used to measure the structural parameters. Environmental effects (such as tidal stripping) may elucidate this result, producing a steeper variation of dark-matter fraction and/or non-homology along the ETG's sequence at higher density. In the optical, the surface brightness slope, b, of the FP increases with local galaxy density, being larger for group relative to field galaxies. The difference vanishes in the NIR, as field galaxies show a small (~2.5 per cent) but significant increase of b from g through K, while group galaxies (particularly those in rich clusters) do not. The trend of b with the environment results from galaxies residing in more massive clusters, since for groups no variation of b with local density is detected. A possible explanation for these findings is that the variation of stellar population properties with mass in ETGs is shallower for galaxies at high density, resulting from tidal stripping and quenching of star formation in galaxies falling into the group's potential well. We do not detect any dependence of the FP coefficients on the presence of substructures in parent galaxy groups.