No.
Take for instance
$Q=x+2x^2+3x^3+4x^4+...+(n-1)x^{n-1}+nx^n+(n-1)x^{n+1}+...+2x^{n-2}+x^{2n-1}$,
for some $n$.
Then let $P=Q(x-1)^2=x^{2n+1}-2x^{n+1}+x$.
In this case, $||P||=2$ but $||Q||$ can be made as large as we want.
Note: it has been proven that cyclotomic polynomials have arbitrarily large coefficients. Cyclotomic polynomials are all factors of $x^n-1$ for some $n$. Therefore, even if $P$ is in the very simple form $x^n-1$, and $||P||=1$, there is no bound on the coefficients of factors of $P$.