There's a bound in Lang's book *Fundamentals of Diophantine Geometry* that's not great in terms of its dependence on the degree and number of variables, but it's very explicit. (Lang works with any field containing $\mathbb{Q}$, but I'll just state it for $\mathbb{C}$ coefficients).

**Proposition 2.3** (page 57) Let $d\ge0$. Let $f$ and $g$ be polynomials in $\mathbb{C}[X_1,\ldots,X_n]$ satisfying $\deg(f)+\deg(g)<d$. Then
$$
\frac{1}{4^{d^n}}|fg|_\infty \le |f|_\infty|g|_\infty \le 4^{d^n}|fg|_\infty.
$$

If one instead uses a $p$-adic absolute value, the same estimate is true without the $4^{d^n}$ factors, i.e., $|fg|_p=|f|_p|g|_p$. This is a version of Gauss's lemma.

Final note: the way that your problem is phrased, there is no bound, since if $G$ is a factor of $F$, then so is $cG$ for any nonzero constant $G$. In Lang's formulation, if you shift a constant from one factor to another, it doesn't affect the statement of the result.