The spectral extrema problems on forbidding minors have aroused wide attention. Very recently, Zhai and Lin [J. Combin. Theory Ser. B 157 (2022) 184--215] determined the extremal graph with maximum adjacency spectral radius among all $K_{s,t}$-minor free graphs of sufficiently large order. The matrix $A_α(G)$ is a generalization of the adjacency matrix $A(G)$, which is defined by Nikiforov \cite{Nikiforov2} as $$A_α(G) = αD(G) + (1 - α)A(G),$$ where $0\leqα\leq1$. Given a graph $F$, the $A_α$-spectral extrema problem is to determine the maximum spectral radius of $A_α(G)$ or characterize the extremal graph among all graphs with no subgraph isomorphic to $F$. For $α=0$, the matrix $A_α(G)$ is exactly the adjacency matrix $A(G)$. Motivated by the nice work of Zhai and Lin, in this paper we determine the extremal graph with maximum $A_α$-spectral radius among all $K_{s,t}$-minor free graphs of sufficiently large order, where $0<α<1$ and $2\leq s\leq t$. As by-products, we completely solve the Conjecture posed by Chen and Zhang in [Linear Multilinear Algebra 69 (10) (2021) 1922--1934].