  A $\textit{square-full}$ number is a positive integer for which all its prime
divisors divide itself at least twice. The counting function of square-full
integers of the form $f(n)$ for $n\leqslant N$ is denoted by
$S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)$. We have known that for a
relatively prime pair $(a,b)\in\mathbb N\times \mathbb N\cup\{0\}$ with a
linear polynomial $f(x)=ax+b$, its counting function is $\asymp_{a,b}
N^\frac{1}{2}$. Fix $\varepsilon>0$, for an admissible quadratic polynomial
$f(x)$, we prove that
$$S^{{\mathstrut\hspace{0.05em}\blacksquare}}_f(N)\ll_{\varepsilon, f}
N^{\varpi+\varepsilon}$$ for some absolute constant $\varpi<1/2$. Under the
assumption on the $abc$ conjecture, we expect the upper bound to be
$O_{\varepsilon,f}(N^\varepsilon)$.
