I would think the answer is yes and follows by applying "[1.0.1] Theorem" on page 2 [here][1] for the finite Borel measures $\Re^+f(g)dg$, $\Re^-f(g)dg$, $\Im^+f(g)dg$, $\Im^-f(g)dg$ on $G$, where $\pm$ stands for positive and negative part. Note that Frechet spaces satisfy the conditions there.

**EDIT.** As the OP pointed out, the result is stated for compactly supported functions. By an approximation argument and the last line of the quoted theorem, an extension seems possible to functions $f:G\to\mathbb{C}$ satisfying $\int_G |f(g)|\ |\pi(g)v|_\mu\ dg < \infty$  for each seminorm $|\cdot|_\mu$ that participates in the definition of the topology of $V$.