The quotient set, or ratio set, of a set of integers $A$ is defined as $R(A)
:= \left\{a/b : a,b \in A,\; b \neq 0\right\}$. We consider the case in which
$A$ is the image of $\mathbb{Z}^+$ under a polynomial $f \in \mathbb{Z}[X]$,
and we give some conditions under which $R(A)$ is dense in $\mathbb{Q}_p$.
Then, we apply these results to determine when $R(S_m^n)$ is dense in
$\mathbb{Q}_p$, where $S_m^n$ is the set of numbers of the form $x_1^n + \cdots
+ x_m^n$, with $x_1, \dots, x_m \geq 0$ integers. This allows us to answer a
question posed in [Garcia et al., $p$-adic quotient sets, Acta Arith. 179,
163-184]. We end leaving an open question