In this work, we deal with the following two-point non-linear Dirichlet boundary value problem for a finite nabla fractional difference equation: \begin{equation*} \begin{cases} -eft(abla^{lpha}_{\rho(a)} u\right)(t) = f(u(t)), \quad t ı \mathbb{N}^{b}_{a + 2}, u(a) = u(b) = 0. \end{cases} \end{equation*} Here $a$, $b \in \mathbb{R}$ with $b - a \in \mathbb{N}_{3}$, $1 < \alpha < 2$, $f :\mathbb{R} \rightarrow \mathbb{R}^{+}\cup\{0\}$ is a continuous function, and $\nabla^{\alpha}_{\rho(a)}$ denotes the $\alpha^{\text{th}}$ order Riemann-Liouville nabla difference operator. First, we construct an associated Green's function and obtain some of its properties. Under suitable conditions on the non-linear part of the difference equation, we deduce some results for at least two and at least three positive solutions of the considered problem. For this purpose, we use a few prominent conical shell fixed point theorems.