According to the Bodmer–Witten hypothesis, ordinary nuclear matter is just a metastable state of the so-called strange quark matter. If it already exists in the universe, it can manifest as quark stars. Since this type of compact star has properties somewhat similar to neutron stars, one might think of them as potential candidates satisfying modern pulsar measurements. In this work, we study the radial and non-radial oscillations of hot and lepton-rich quark stars, i.e., instants after their potential birth in supernovae events. In order to calculate their fundamental-mode frequencies, we use a density-dependent quark mass framework for microphysics to build the equation of state, solve the eigenvalue problem to determine their radial stability, and use the full general-relativistic perturbative framework of non-radial oscillations. We found that their dependencies on the gravitational masses, compactness, and energy densities display remarkable differences compared to widely studied protoneutron stars. In order to quantify these differences, we build and propose some universal relations correlating all these stellar parameters. As expected, similar hadronic universal relations depart from our results. In this sense, our findings may serve as a smoking gun to distinguish strange quark stars from ordinary nuclear neutron stars using upcoming gravitational-wave data in the multimessenger astronomy era.