Let $n$ be a positive integer, $g\in H(\mathbb{D})$ and $\varphi$ be an analytic self-map of $\mathbb{D}$. The boundedness and compactness of the integral operator $$ (C^n_{\varphi,g}f)(z)=\int_{0}^{z}f^{(n)}(\varphi(\xi))g(\xi)d\xi $$ from the Bloch and little Bloch space into the spaces $Q_K(p,q)$ and $Q_{K,0}(p,q)$ are characterized.