Since $f$ is a morphism, $(xy)^{m+1} = x^{m+1}y^{m+1}$, so $(yx)^m = x^m y^m$ for all $x$ and all $y$ in $G$. Further, $y^{m+1}x^{m+1} = (yx)^{m+1} = (yx)^m(yx) = (x^m y^m)(yx) = x^m y^{m+1}x$, so $y^{m+1}x^m = x^m y^{m+1}$. Since $f$ is surjective, the latter shows that $x^m$ is in the center of $G$. Similarly, $x^n$ is in the center of $G$. Finally, since $m$ is coprime to $n$, it follows that every element of $G$ is in the center, so $G$ is commutative.