In this paper, we study the maximum \(\alpha \) -quasi-clique (MQC) problem, which seeks the largest subgraph with edge density at least \(\alpha \) , where \(\alpha \in (0,1]\) is a user-specified parameter. Quasi-cliques generalize cliques, with 1-quasi-cliques corresponding exactly to cliques. The MQC problem is NP-hard, and existing heuristic algorithms such as \(\textsf{NB}\) , \(\textsf{NBSim}\) and \(\textsf{FastNBSim}\) suffer from two major limitations: (i) their solution quality is often suboptimal, and (ii) they must rerun costly algorithms to discovery alternative dense subgraphs for different \(\alpha \) values. We propose an index-based approach for efficiently querying large quasi-cliques. Our index stores size-density pairs \(\{(k,\rho _k) \mid k \in [2,n]\}\) together with their corresponding subgraphs, where \(\rho _k\) is the edge density of a k-vertex subgraph obtained by our index construction algorithms. Given a query parameter \(\alpha \) , we return the largest k with \(\rho _k \ge \alpha \) along with its associated subgraph. We introduce two index construction algorithms, \(\textsf{Peeling}\) and \(\textsf{NeiPeeling}\) , which run in \( \mathcal{O}(m+n)\) and \( \mathcal{O}(\delta (G)\cdot m)\) time, respectively, and produce indexes of size \( \mathcal{O}(n)\)  and \( \mathcal{O}(m+n)\) . Both indexes satisfy the monotonicity property, enabling query processing in time linear in the result size. Extensive experiments on nine large real-world graphs demonstrate the efficiency and effectiveness of our algorithms.