We study properties of Keplerian disks and their high-frequency quasi-periodic oscillations (HF QPOs) in the field of quark stars with dimensionless spin a breaking the black-hole spin limit of a = 1 up to a approximate to 1.3. Using the external geometry of the superspinning quark stars approximated by the Kerr geometry, we show that the Keplerian disks have to touch the surface of such quark stars and their accretion efficiency eta approximate to 18% significantly exceeds the efficiency related to the Schwarzschild black holes. Using the geodesic oscillation models, we test possible existence of the superspinning quark stars in atoll sources demonstrating the twin HF QPOs with resonant frequency ratios 3:2, 4:3, 5:4. For explanation of the twin HF QPOs we consider the standard relativistic precession model and its modifications, the tidal distortion model, the resonance epicyclic and the warped disk model. In a given model, we assume occurrence of the twin oscillatory modes at a common resonant dimensionless radius x = r/M determined by the frequency ratio and the quark star spin a. The theoretical limit R > 3M on the quark star surface radius puts strong restrictions on the relations between the resonant radii x and the quark star spin a. These restrictions imply that all the considered geodesic oscillation models can be excluded, except for one variant of the relativistic precession model, or alternatively the tidal distortion and warped disk models, that allow for appearance of the twin HF QPOs with frequency ratio 3 : 2 at radii slightly above the theoretical limit on the radius of the quark star surface, but exclude the smaller frequency ratios (4:3, 5:4).