Let $X$ be an integral (and singular) curve over the complex field and let $A$ be a torsion free sheaf (not necessarily locally free) of rank $1$ on $X$. We denote by $\mathrm{Quot}^n_A$ the Quot scheme parametrizing quotient $q:A\to Q$ of length $n$. Is this Quot scheme always isomorphic to the Hilbert scheme $\mathrm{Hilb}^n(X)$? If not, can you give me a counterexample?

That is definitely not true. Let $X$ be an integral curve that has a single ordinary double point $p$ (and no other singular points). Let $\nu:\widetilde{X}\to X$ be the normalization. The fiber of $\nu$ over $p$ consists of two closed points, $p_1$ and $p_2$. Let $A$ be $\nu_*\mathcal{O}_{\widetilde{X}}$. Let $n$ be $1$. Then $\text{Quot}^1_A$ admits a "fundamental cycle" morphism, $$\text{FC}:\text{Quot}^1_A \to X.$$ 

