No. For example, Robinson’s Q + $\forall x\,G_y\,x<y$ is a categorical theory (its only model up to isomorphism is the standard model of arithmetic), hence it is not equivalent to any first-order sentence (or theory).
EDIT: In order to avoid confusion: my answer above assumes that “every countable model of natural numbers” in the OP is interpreted so that it includes at least one nonstandard model of true arithmetic. The Marker and Slaman paper mentioned in the comments seemingly contradicts what I wrote as they call the almost-all quantifier to be a fragment of first-order logic. The explanation is that they only care about validity in the standard model $\mathbb N$: then $G_x\,\phi(x)$ is equivalent to the first-order formula $\exists u\,\forall x\,(u\le x\to\phi(x))$. However, this equivalence is not valid in any nonstandard model of arithmetic.