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 $dot z eq 0$, 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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