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The lead-free halide double perovskite Cs₂InAgCl₆ possesses a wide direct band-gap exceeding 3 eV, and exhibits very broad and strong photoluminescence (PL) in the visible region [G. Volonakis, et al.,  J. Phys. Chem. Lett. 8, 772 (2017)]. This property has been employed to realize white-light-emitting diodes with near-unity PL quantum yield [J. Luo, et al., Nature 563, 541 (2018); S. Li et al., ACS Appl. Mater. Interfaces 12, 46330 (2020)]. The atomic scale mechanisms of light emission in this compound remain a subject of debate, with proposals ranging from self-trapped excitons to defect-assisted luminescence. To shed light on this debate, in previous work [V.-A Ha, et al., J. Phys. Chem. C 125, 21689 (2021)] we investigated the electronic band structure of Cs₂InAgCl₆ using GW method, and the temperature-dependent band gap renormalization via electron-phonon interactions within special displacement method. Here, we extend this study to include excitonic effect via the Bethe-Salpeter equation, and their coupling to phonons via the special displacement method. We will discuss our results in relation to the observed PL signal.