We analyze existing measurements of [Fe/H] and [$alpha$/Fe] for individual red giant branch (RGB) stars in the Giant Stellar Stream (GSS) of M31 to determine whether spatial abundance gradients are present. These measurements were obtained from low- ($R sim 3000$) and moderate- ($R sim 6000$) resolution Keck/DEIMOS spectroscopy using spectral synthesis techniques as part of the Elemental Abundances in M31 survey. From a sample of 62 RGB stars spanning the GSS at 17, 22, and 33 projected kpc, we measure a [Fe/H] gradient of $-$0.018 $pm$ 0.003 dex kpc$^{-1}$ and negligible [$alpha$/Fe] gradient with M31-centric radius. We investigate GSS abundance patterns in the outer halo using additional [Fe/H] and [$alpha$/Fe] measurements for 6 RGB stars located along the stream at 45 and 58 projected kpc. These abundances provide tentative evidence that the trends in [Fe/H] and [$alpha$/Fe] beyond 40 kpc in the GSS are consistent with those within 33 kpc. We also compare the GSS abundances to 65 RGB stars located along the possibly related Southeast (SE) shelf substructure at 12 and 18 projected kpc. The abundances of the GSS and SE shelf are consistent, supporting a common origin hypothesis, although this interpretation may be complicated by the presence of [Fe/H] gradients in the GSS. We discuss the abundance patterns in the context of photometric studies from the literature and explore implications for the properties of the GSS progenitor, suggesting that the high $langle$[$alpha$/Fe]$rangle$ of the GSS (+0.40 $pm$ 0.05 dex) favors a major merger scenario for its formation.
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