Let $A(G)$ and $D(G)$ be the adjacency and degree matrices of a simple graph $G$ on $n$ vertices, respectively. The \emph{$A_α$-spectral radius} of $G$ is the largest eigenvalue of $A_α(G)=αD(G)+(1-α)A(G)$ for a real number $α\in[0,1]$. In this paper, for $α\in (0,1)$, we obtain a sharp upper bound for the $A_α$-spectral radius of graphs on $n$ vertices without a subgraph isomorphic to a liner forest for $n$ large enough and characterize all graphs which attain the upper bound. As a result, we completely obtain the maximum signless Laplacian spectral radius of graphs on $n$ vertices without a subgraph isomorphic to a liner forest for $n$ large enough.