Searches for the exclusive decays of the Higgs boson to an 𝜔 meson and a photon or a 𝐾* meson and a photon can probe flavour-conserving and flavour-violating Higgs boson couplings to light quarks, respectively. Searches for these decays, along with the analogous 𝑍 boson decay to an 𝜔 meson and a photon, are performed with a 𝑝𝑝 collision data sample corresponding to integrated luminosities of up to 134 fb⁻¹ collected at √𝑠 = 13 TeV with the ATLAS detector at the CERN Large Hadron Collider. The obtained 95% confidence-level upper limits on the respective branching fractions are (𝐻 → 𝜔𝛾) < 5.5 × 10⁻⁴, (𝐻 → 𝐾*𝛾) < 2.2 × 10⁻⁴ and (𝑍 → 𝜔𝛾) < 3.9 × 10⁻⁶. The limits for 𝐻 → 𝜔𝛾 and 𝑍 → 𝜔𝛾 are 370 times and 140 times the Standard Model expected values, respectively. The result for 𝑍 → 𝜔𝛾 corresponds to a two-orders-of-magnitude improvement over the limit obtained by the DELPHI experiment at LEP.