For a connected graph $G$, let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal matrix of the degrees of the vertices in $G$. The $A_α$-matrix of $G$ is defined as \begin{align*} A_α(G) = αD(G) + (1-α) A(G) \quad \text{for any $α\in [0,1]$}. \end{align*} The largest eigenvalue of $A_α(G)$ is called the $A_α$-spectral radius of $G$. In this article, we characterize the graphs with maximum $A_α$-spectral radius among the class of unicyclic and bicyclic graphs of order $n$ with fixed girth $g$. Also, we identify the unique graphs with maximum $A_α$-spectral radius among the class of unicyclic and bicyclic graphs of order $n$ with $k$ pendant vertices.