  We present analysis of the mass-metallicity relation (MZR) for a sample of 67
[OIII]-selected star-forming galaxies at a redshift range of $z=1.99 - 2.32$
($z_{\text{med}} = 2.16$) using \emph{Hubble Space Telescope} Wide Field Camera
3 grism spectroscopy from the Quasar Sightline and Galaxy Evolution (QSAGE)
survey. Metallicities were determined using empirical gas-phase metallicity
calibrations based on the strong emission lines [OII]3727,3729, [OIII]4959,5007
and H$\beta$. Star-forming galaxies were identified, and distinguished from
active-galactic nuclei, via Mass-Excitation diagrams. Using $z\sim0$
metallicity calibrations, we observe a negative offset in the $z=2.2$ MZR of
$\approx -0.51$ dex in metallicity when compared to locally derived
relationships, in agreement with previous literature analysis. A similar offset
of $\approx -0.46$ dex in metallicity is found when using empirical metallicity
calibrations that are suitable out to $z\sim5$, though our $z=2.2$ MZR, in this
case, has a shallower slope. We find agreement between our MZR and those
predicted from various galaxy evolution models and simulations. Additionally,
we explore the extended fundamental metallicity relation (FMR) which includes
an additional dependence on star formation rate (SFR). Our results consistently
support the existence of the FMR, as well as revealing an offset of
$0.28\pm0.04$ dex in metallicity compared to locally-derived relationships,
consistent with previous studies at similar redshifts. We interpret the
negative correlation with SFR at fixed mass, inferred from an FMR existing for
our sample, as being caused by the efficient accretion of metal-poor gas
fuelling SFR at cosmic noon.
