Cohesive subgraph mining is a fundamental problem in graph theory with numerous real-world applications, such as social network analysis and protein-protein interaction modeling. Among various cohesive subgraphs, the $γ$-quasi-clique is widely studied for its flexibility in requiring each vertex to connect to at least a $γ$ proportion of other vertices in the subgraph. However, solving the maximum $γ$-quasi-clique problem is NP-hard and further complicated by the lack of the hereditary property, which makes designing efficient pruning strategies challenging. Existing algorithms, such as DDA and FastQC, either struggle with scalability or exhibit significant performance declines for small values of $γ$. In this paper, we propose a novel algorithm, IterQC, which reformulates the maximum $γ$-quasi-clique problem as a series of $k$-plex problems that possess the hereditary property. IterQC introduces a non-trivial iterative framework and incorporates two key optimization techniques: (1) the pseudo lower bound (pseudo LB) technique, which leverages information across iterations to improve the efficiency of branch-and-bound searches, and (2) the preprocessing technique that reduces problem size and unnecessary iterations. Extensive experiments demonstrate that IterQC achieves up to four orders of magnitude speedup and solves significantly more graph instances compared to state-of-the-art algorithms DDA and FastQC.