We present here quantitative diagnostic tools for cool giants that employ low-resolution near-infrared spectroscopy in the $K$-band for stellar population studies. In this study, a total of 260 cool giants (177 stars observed with X-shooter and 83 stars observed with NIFS) are used covering a wider metallicity range than in earlier works. We measure equivalent widths of some of the selected important $K$-band spectral features like Na I, Fe I, and $^{12}$CO after degrading the spectral resolution (R $sim$ 1200) to investigate the spectral behavior with fundamental parameters (e.g. effective temperature and metallicity). We derive empirical relations to measure effective temperature using the $^{12}$CO first-overtone band at 2.29 $mu$m and 2.32 $mu$m and show a detailed quantitative metallicity dependence of these correlations. We find that the empirical relations based on solar-neighborhood stars can incorporate large uncertainty in evaluating $T_{eff}$ for metal-poor or metal-rich stars. Furthermore, we explore all the spectral lines to establish the empirical relation with metallicity and find that the quadratic fit of the combination of Na I and $^{12}$CO at 2.29 $mu$m lines yields a reliable empirical relation at [$Fe/H$] $leq$ $-$0.4 dex, while a linear fit of any line offers a good metallicity scale for stars having [$Fe/H$] $geq$ 0.0 dex.
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