The Galactic Centre (GC) has experienced a high degree of recent star-forming activity, as evidenced by the large number of massive stars currently residing there. The relative abundances of chemical elements in the GC may provide insights into the origins of this activity. Here, we present high-resolution H-band spectra of two Red Supergiants in the GC (IRS 7 and VR 5-7), and in combination with spectral synthesis we derive abundances for Fe and C, as well as other alpha -elements Ca, Si, Mg Ti and O. We find that the C-depletion in VR 5-7 is consistent with the predictions of evolutionary models of RSGs, while the heavy depletion of C and O in IRS 7's atmosphere is indicative of deep mixing, possibly due to fast initial rotation and/or enhanced mass-loss. Our results indicate that the current surface Fe/H content of each star is slightly above Solar. However, comparisons to evolutionary models indicate that the initial Fe/H ratio was likely closer to Solar, and has been driven higher by H-depletion at the stars' surface. Overall, we find alpha /Fe ratios for both stars which are consistent with the thin Galactic disk. These results are consistent with other chemical studies of the GC, given the precision to which abundances can currently be determined. We argue that the GC abundances are consistent with a scenario in which the recent star-forming activity in the GC was fuelled by either material travelling down the Bar from the inner disk, or from the winds of stars in the inner Bulge - with no need to invoke top-heavy stellar Initial Mass Functions to explain anomalous abundance ratios.