No.  Let $X = E \times E$ with $E$ an elliptic curve and let $G = \mathbb Z_2 \oplus \mathbb Z_2$, with each factor acting on one of the $E$'s by the involution and fixing the other.  The quotient is $\mathbb P^1 \times \mathbb P^1$.  The effective cone of $E \times E$ is round, while the effective cone of $\mathbb P^1 \times \mathbb P^1$ is polyhedral.

The converse seems true, though: if $X$ has polyhedral effective cone, then so does $Y$, spanned by the pushforwards of the generators of the cone for $X$.