We present the first VLT near-IR observations of a gravitationally lensed quasar, using adaptive optics and laser guide star. These observations can be considered as a test bench for future systematic observations of lensed quasars with adaptive optics, even when bright natural guide stars are not available in the nearby field. With only 14 min of observing time, we derived very accurate astrometry of the quasar images and of the lensing galaxy, with 0.05 '' spatial resolution, comparable to the Hubble Space Telescope (HST). In combination with deep VLT optical spectra of the quasar images, we use our adaptive optics images to constrain simple models for the mass distribution of the lensing galaxy. The latter is almost circular and does not need any strong external shear to fit the data. The time delay predicted for SDSS J0806+2006, assuming a singular isothermal ellipsoid model and the concordance cosmology, is Delta t similar or equal to 50 days. Our optical spectra indicate a flux ratio between the quasar images of A/B = 1.3 in the continuum and A/B = 2.2 in both the Mg II and in the C III] broad emission lines. This suggests that microlensing affects the continuum emission. However, the constant ratio between the two emission lines indicates that the broad emission line region is not microlensed. Finally, we see no evidence of reddening by dust in the lensing galaxy.