We study the astrophysical implications of the quark nugget model of dark matter and propose observational techniques for detecting antiquark nuggets (anti-QNs) with modern telescopes. Anti-QNs are compact composite objects of antiquark matter with a typical radius $R\ensuremath{\sim}{10}^{\ensuremath{-}5}\text{ }\text{ }\mathrm{cm}$ and density exceeding that of nuclear matter. Atoms and molecules of interstellar medium collide with antiquark nuggets and annihilate. We estimate thermal radiation from anti-QNs in cold molecular clouds in our Galaxy and show that this radiation appears sufficiently strong to be observed in infrared and visible spectra. Proton annihilation on anti-QNs produces $\ensuremath{\gamma}$ photons with energies in the range 100--400 MeV, which may be detected by telescopes such as Fermi-LAT. We have found that anti-QN radiation inside the solar corona is too weak to produce a significant plasma heating or any other observable effects, while the radiation of $\ensuremath{\gamma}$ photons from the chromosphere may be observable. We also address the problem of survival of antiquark nuggets in the early Universe.