From high resolution (R similar or equal to 45,000), high signal-to-noise ratio (S/N > 400) spectra gathered with the Immersion Grating Infrared Spectrograph (IGRINS) in the H and K photometric bands, we have derived elemental abundances of two bright, well-known metal-poor halo stars: the red giant HD 122563 and the subgiant HD 140283. Since these stars have metallicities approaching [Fe/H] = -3, their absorption features are generally very weak. Neutral-species lines of Mg, Si, S and Ca are detectable, as well as those of the light odd-Z elements Na and Al. The derived IR-based abundances agree with those obtained from optical-wavelength spectra. For Mg and Si the abundances from the infrared transitions are improvements to those derived from shorter wavelength data. Many useful OH and CO lines can be detected in the IGRINS HD 122563 spectrum, from which derived O and C abundances are consistent to those obtained from the traditional [O I] and CH features. IGRINS high resolutions Hand K-band spectroscopy offers promising ways to determine more reliable abundances for additional metal-poor stars whose optical features are either not detectable, or too weak, or are based on lines with analytical difficulties.