Gelfand's inequality is probably what you want here; see for example my book with Hindry, Diophantine Geometry: An Introduction, Proposition B.7.3. I'll state it for polynomials in $\mathbb{Z}[X_1,\ldots,X_m]$, although there's a version that's true over $\overline{\mathbb{Q}}$. The statement uses the projective height, so for a polynomial $f$ with coefficients $a_i\in\mathbb{Z}$, we let $$H(f) = \frac{\max|a_i|}{\gcd(a_i)}.$$ Then

Proposition B.7.3 (Gelfand's inequality) Let $f_1,\ldots,f_r\in \mathbb{Z}[X_1,\ldots,X_m]$, and for $1\le i\le m$, let $d_i$ denote the $X_i$ degree of $f_1f_2\cdots f_r$. Then $$ H(f_1)H(f_2)\cdots H(f_r) \le e^{d_1+\cdots+d_m}H(f_1f_2\cdots f_r). $$

For the OP's question, we have $f$ is divisible by $g$, say $f=gg'$, so $$ H(g) \le H(g)H(g') \le e^{\deg f}H(gg') = e^{\deg f}H(f). $$

Gelfond's inequality is probably what you want here; see for example my book with Hindry, Diophantine Geometry: An Introduction, Proposition B.7.3. I'll state it for polynomials in $\mathbb{Z}[X_1,\ldots,X_m]$, although there's a version that's true over $\overline{\mathbb{Q}}$. The statement uses the projective height, so for a polynomial $f$ with coefficients $a_i\in\mathbb{Z}$, we let $$H(f) = \frac{\max|a_i|}{\gcd(a_i)}.$$ Then

Proposition B.7.3 (Gelfond's inequality) Let $f_1,\ldots,f_r\in \mathbb{Z}[X_1,\ldots,X_m]$, and for $1\le i\le m$, let $d_i$ denote the $X_i$ degree of $f_1f_2\cdots f_r$. Then $$ H(f_1)H(f_2)\cdots H(f_r) \le e^{d_1+\cdots+d_m}H(f_1f_2\cdots f_r). $$

For the OP's question, we have $f$ is divisible by $g$, say $f=gg'$, so $$ H(g) \le H(g)H(g') \le e^{\deg f}H(gg') = e^{\deg f}H(f). $$