We report the first detailed chemical abundance analysis of the exoplanet-hosting M-dwarf stars Kepler-138 and Kepler-186 from the analysis of high-resolution ($R$ $∼$ 22,500) $H$-band spectra from the SDSS IV - APOGEE survey. Chemical abundances of thirteen elements - C, O, Na, Mg, Al, Si, K, Ca, Ti, V, Cr, Mn, and Fe - are extracted from the APOGEE spectra of these early M-dwarfs via spectrum syntheses computed with an improved line list that takes into account H$_{2}$O and FeH lines. This paper demonstrates that APOGEE spectra can be analyzed to determine detailed chemical compositions of M-dwarfs. Both exoplanet-hosting M-dwarfs display modest sub-solar metallicities: [Fe/H]$_{Kepler-138}$ = -0.09 $±$ 0.09 dex and [Fe/H]$_{Kepler-186}$ = -0.08 $±$ 0.10 dex. The measured metallicities resulting from this high-resolution analysis are found to be higher by $∼$0.1-0.2 dex than previous estimates from lower-resolution spectra. The C/O ratios obtained for the two planet-hosting stars are near-solar, with values of 0.55 $±$ 0.10 for Kepler-138 and 0.52 $±$ 0.12 for Kepler-186. Kepler-186 exhibits a marginally enhanced [Si/Fe] ratio. ; Comment: 28 pages, 5 figures, 4 tables. Accepted to ApJ