Using density-functional theory, we calculate the electronic band structure of single-layer graphene on top of hexagonal In2Te2 monolayers. The geometric configuration with In and Te atoms at the centers of the carbon hexagons leads to a Kekulé texture with an ensuing band gap of 20 meV. The alternative structure, nearly degenerate in energy, with the In and Te atoms on top of carbon sites, is characterized instead by a gapless spectrum with the original Dirac cones of graphene reshaped, depending on the graphene-indium chalcogenide distance, either in the form of an undoubled pseudospin-one Dirac cone or in a quadratic band crossing point at the Fermi level. These electronic phases harbor charge fractionalization and topological Mott insulating states of matter.