In this paper, we study the BGG category $\mathcal{ O}$ for the quantum Schr{ \"o}dinger algebra $U_q(\mathfrak{ s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $\dot z\neq 0$, using the module $B_{ \dot z}$ over the quantum Weyl algebra $H_q$, we show that there is an equivalence between the full subcategory $\mathcal{ O}[\dot z]$ consisting of modules with the central charge $\dot z$ and the BGG category $\mathcal{ O}^{ (\mathfrak{ sl}_2)}$ for the quantum group $U_q(\mathfrak{ sl}_2)$. In the case that $\dot z=0$, we study the subcategory $\mathcal{ A}$ consisting of finite dimensional $U_q(\mathfrak{ s})$-modules of type $1$ with zero action of $Z$. Motivated by the ideas in \cite{ DLMZ, Mak}, we directly construct an equivalent functor from $\mathcal{ A}$ to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional $U_q(\mathfrak{ s})$-modules is wild.
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