First-principles calculations were carried out by the Vienna Ab initio Simulation Package (VASP). Preliminary calculations showed that DFT-D3 method with Becke-Jonson damping present good agreement of lattice constant with experiment data. Ground state of graphene position on the SiO2 surfaces obtained has been determined. Interlayer distances between graphene and different types of quartz substrate have been calculated; the interlayer distances being 3.31 Å and 4.32 Å for models with open oxygen dangling bonds and with open silicon dangling bonds, respectively. The adsorption energy of graphene on the amorphous SiO2 surface with open oxygen dangling bonds is larger than the adsorption energy on the second type of surface. We observed the 0.12 eV band gap in the case with open oxygen dangling bonds. This kind of quartz surface can be source of a charge puddle.