In this paper, we study composition operators on Bloch space and $Q_K(p,q)$ spaces. We give a Carleson measure characterization on $Q_K(p,q)$ spaces, then we use this Carleson measure characterization of the compact compositions on $Q_K(p,q)$ spaces to show that every compact composition operator on $Q_K(p,q)$ spaces is compact on Bloch space. Moreover, necessary and sufficient condition for $C_\phi$ from the Bloch space $\mathcal B$ to a general class of analytic functions $Q_K(p,q)$ to be compact is given in terms of the map $\phi$.