We present adaptive-optics assisted near-infrared high-spectral-resolution observations of late-type giants in the nuclear star cluster of the Milky Way. The metallicity and elemental abundance measurements of these stars offer us an opportunity to understand the formation and evolution of the nuclear star cluster. In addition, their proximity to the supermassive black hole (∼0.5 pc) offers a unique probe of the star formation and chemical enrichment in this extreme environment. We observed two stars identified by medium spectral-resolution observations as potentially having very high metallicities. We use spectral-template fitting with the PHOENIX grid and Bayesian inference to simultaneously constrain the overall metallicity, [M/H], alpha-element abundance [α/Fe], effective temperature, and surface gravity of these stars. We find that one of the stars has very high metallicity ([M/H] > 0.6) and the other is slightly above solar metallicity. Both Galactic center stars have lines from scandium (Sc), vanadium (V), and yttrium (Y) that are much stronger than allowed by the PHOENIX grid. We find, using the spectral synthesis code Spectroscopy Made Easy, that [Sc/Fe] may be an order of magnitude above solar. For comparison, we also observed an empirical calibrator in NGC 6791, the highest metallicity cluster known ([M/H] ∼ 0.4). Most lines are well matched between the calibrator and the Galactic center stars, except for Sc, V, and Y, which confirms that their abundances must be anomalously high in these stars. These unusual abundances, which may be a unique signature of nuclear star clusters, offer an opportunity to test models of chemical enrichment in this region.