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Neutrinoless Double-Beta decay (0$\nu\beta\beta$) processes sample a wide range of intermediate forbidden nuclear transitions, which may be impacted by quenching of the axial vector coupling constant ($g_A/g_V$), the uncertainty of which plays a pivotal role in determining the sensitivity reach of 0$\nu\beta\beta$ experiments. In this Letter, we present measurements performed on a high-resolution LiInSe$_{2}$~ bolometer in a ''source=detector'' configuration to measure the spectral shape of the 4-fold forbidden $\beta$-decay of $^{115}$In. The value of $g_A/g_V$ is determined by comparing the spectral shape of theoretical predictions to the experimental $\beta$ spectrum taking into account various simulated background components as well as a variety of detector effects. We find evidence of quenching of $g_A/g_V$ at $>5\sigma$ with a model-dependent quenching factor of $0.655\pm0.002$ as compared to the free-nucleon value for the Interacting Shell Model. We also measured the $^{115}$In half-life to be [$5.18\pm0.06(\text{stat.})^{+0.005}_{-0.015}(\text{sys.})]\times{10}^{14}$ yr within the Interacting Shell Model framework. This work demonstrates the power of the bolometeric technique to perform precision nuclear physics single-$\beta$ decay measurements, which can help reduce the uncertainties in the calculation of $0\nu\beta\beta$ nuclear matrix elements.