Hydrogenation of graphene on the α-quartz (0001) SiO_2 substrate is studied, considering different surface terminations in order to take into account the amorphic nature of the material. Our ab initio calculations show that the formation of graphane by hydrogen adsorption on graphene is energetically favored on hydroxyl and oxygen terminated surfaces, whereas silicon termination and reconstruction of the oxygen termination hinder adsorption. Our results indicate that in order to fabricate graphane on SiO_2, it is beneficial to oxygenize the surface and saturate it with hydrogen. For the pristine graphene on the substrate, we find only marginal changes in the low-energy band structure for all surface terminations.