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G1, also known as Mayall II, is one of the most massive star clusters in M31. Its mass, ellipticity, and location in the outer halo make it a compelling candidate for a former nuclear star cluster. This paper presents an integrated light abundance analysis of G1, based on a moderately high-resolution (R=15,000) spectrum obtained with the High Resolution Spectrograph on the Hobby-Eberly Telescope in 2007 and 2008. To independently determine the metallicity, a moderate resolution (R~4,000) spectrum of the calcium-II triplet lines in the near-infrared was also obtained with the Astrophysical Research Consortium's 3.5-m telescope at Apache Point Observatory. From the high-resolution spectrum, G1 is found to be a moderately metal-poor cluster, with [Fe/H]=-0.98+/-0.05. G1 also shows signs of alpha-enhancement (based on Mg, Ca, and Ti) and lacks the s-process enhancements seen in dwarf galaxies (based on comparisons of Y, Ba, and Eu), indicating that it originated in a fairly massive galaxy. Intriguingly, G1 also exhibits signs of Na and Al enhancement, a unique signature of GCs -- this suggests that G1's formation is intimately connected with GC formation. G1's high [Na/Fe] also extends previous trends with cluster velocity dispersion to an even higher mass regime, implying that higher mass clusters are more able to retain Na-enhanced ejecta. The effects of intracluster abundance spreads are discussed in a subsequent paper. Ultimately, G1's chemical properties are found to resemble other M31 GCs, though it also shares some similarities with extragalactic nuclear star clusters.