For a graph $G$ and a real number $\alpha\in [0,1],$ Nikiforov (2017) proposed the $A_\alpha$-matrix for $G,$ which is defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The largest eigenvalue of $A_{\alpha}(G)$ is called the $A_\alpha$-index of $G.$ Let $\mathcal{F}$ be a set of graphs, we say a graph $G$ is $\mathcal{F}$-free if it does not contain a member in $\mathcal{F}$ as a subgraph. In 2010, Nikiforov conjectured that for $n$ large enough, the $\{C_{2k+1}, C_{2k+2}\}$-free graph of maximum spectral radius is $S_{n,k}$, the join of a clique on $k$ vertices with an independent set of $n-k$ vertices and that the $C_{2k+2}$-free graph of maximum spectral radius is $S_{n,k}^+$, the graph obtained from $S_{n,k}$ by adding one edge. Cioabă, Desai and Tait (2022) used a novel method to solve this two-part conjecture. We also note that the well-known Mantel's theorem, which claims that every graph of order $n$ with size $m>\lfloor n/4\rfloor$ contains a triangle. Recently, Zhai and Shu (2022) obtained a spectral version of Mantel's theorem. In this paper, on the one hand, with the aid of Cioabă-Desai-Tait's novel method, we identify the graphs with the first two largest $A_\alpha$-indices among the $n$-vertex $C_4$-free graphs for $0<\alpha<1$ and $n\geq \frac{9}{\alpha^6}$. On the other hand, with the help of Zhai-Shu's eigenvector method, we identify the $C_4$-free graphs (other than the star) of size $m$ with no isolated vertex having the largest $A_\alpha$-index for $\frac{1}{2}\leq\alpha<1$ and $m\geq 3$. Our results improve some known ones of Tian, Chen, Cui (2021), Guo, Zhang (2022), Feng, Wei (2022) and Li, Qin (2021).
