Let $p(n)$ and $q(n)$ by polynomials with integer coefficients such that $p(n)|q(n)$ for infinitely many integers $n$. Is there a polynomial $r(n)$ such that $q(n)=p(n)r(n)$?
Note that this is not true if we require $r(n)$ to be integer-valued: $q(n)=n$ and $p(n)=p$ for some prime $p$ provides a counterexample.
Write $dq = pr+s$ where $s,r \in \Bbb Z[X]$, $\deg s < \deg p$ and $d$ is a (nonzero) positive integer. (this is always possible by picking for $d$ the dominant coefficient of $p$ to the $(\deg q - \deg p+1)$th power)
Your hypothesis implies that $p(n)$ divides $s(n)$ for infinitely many $n$.
For $n$ large enough, $|p(n)| > |s(n)|$ so you must have $s(n)=0$ for infinitely many $n$, hence $s=0$.
Then $q = pr'$ where $r'=r/d$ is a polynomial with rational coefficients.