In this study, we construct the difference sequence spaces $\ell_p(\mathcal P\nabla^2_q)=(\ell_p)_{\mathcal P\nabla^2_1}$, $1\leq p\leq\infty$, where $\mathcal P=\varrho_{rs}$ is an infinite matrix of Padovan numbers $\varrho_s$ defined by \[ ǎrrho_{rs}= \begin{cases} \frac{ǎrrho_s}{ǎrrho_{r+5}-2}, 0eq seq r 0, s>r \end{cases}. \] and $\nabla^2_q$ is a $q$-difference operator of second order. We obtain some inclusion relations, topological properties, Schauder basis and $\alpha$-, $\beta$- and $\gamma$-duals of the newly defined space. We characterize certain matrix classes from the space $\ell_p(\mathcal{P}\nabla^2_q)$ to any one of the space $\ell_1$, $c_0$, $c$ or $\ell_\infty$. We examine some geometric properties and give certain estimation for von-Neumann Jordan constant and James constant of the space $\ell_p(\mathcal{P})$. Finally, we estimate upper bound for Hausdorff matrix as a mapping from $\ell_p$ to$\ell_p(\mathcal{P})$.