We study the hadron-quark phase transition in the interior of neutron stars. For the hadronic sector, we use a microscopic equation of state involving nucleons and hyperons derived within the Brueckner-HartreeFock many-body theory with realistic two-body and three-body forces. For the description of quark matter, we employ the Dyson-Schwinger approach and compare with the MIT bag model. We calculate the structure of neutron star interiors comprising both phases and find that with the Dyson-Schwinger model, the hadron-quark phase transition takes place only when hyperons are excluded, and that a two-solar-mass hybrid star is possible only if the nucleonic equation of state is stiff enough.