$\newcommand\si\sigma$Yes. 

Indeed, 
$$F_p:=F_p(m,\si)=\|(\si-1)Z+mR\|_p,$$
where $Z$ and $R$ are independent random variables, $Z$ is standard normal, $R$ is Rademacher (with $P(R=\pm1)=1/2$), and $\|X\|_p:=(E|X|^p)^{1/p}$. 

So, the desired result follows by the central limit theorem and the [Haagerup inequalities][1], which imply that, for each real $p>0$,
$$G_p:=\Big\|\sum_{i=1}^n a_iR_i\Big\|_p$$
differs from $G_2$ by at most a factor depending only on $p$, where the $a_i$'s are any nonzero real numbers and the $R_i$'s are independent Rademacher r.v.'s.

  [1]: https://eudml.org/doc/218383