The answer to your question as stated is no.
What discrete Morse theory gives you, starting from a finite regular CW complex $X$ and a discrete Morse function $f:X \to \mathbb{R}$ (with discrete vector field $V$) is a chain complex generated by the critical cells
$$ \cdots \stackrel{\partial_{k+1}}{\longrightarrow} M_k \stackrel{\partial_k}{\longrightarrow} M_{k-1} \stackrel{\partial_{k-1}}{\longrightarrow} \cdots \stackrel{\partial_1}{\longrightarrow} M_0$$
whose homology is isomorphic to that of $X$ via a chain map $\phi:(X,d) \to (M,\partial)$ which is given by $\phi = (1 + dV + Vd)^N$ for some suitably huge $N$. Now if all the $\partial_*$ are trivial (i.e., if you have a so-called perfect Morse function), then all is well: the homology generators here are just the chains themselves, and in Section 7 of the original paper
R Forman, Morse theory for cell complexes, Adv. Math 90-145, 1998
Forman shows that $M_k$ consists precisely of the $\phi$-invariant $k$-dimensional chains in $X$. There is no direct recipe, as far as I'm aware, of generating equivalence classes of $\phi$-invariant chains which represent the same integral homology class in $X$ when the boundary operators $\partial_*$ end up being nontrivial.