The $ k $-plex model, which allows each vertex to miss connections with up to $ k $ neighbors, serves as a relaxation of the clique. Its adaptability makes it more suitable for analyzing real-world graphs where noise and imperfect data are common and the ideal clique model is often impractical. The problem of identifying the maximum $ k $-plex (MKP, which is NP-hard) is gaining attention in fields such as social network analysis, community detection, terrorist network identification, and graph clustering. Recent works have focused on optimizing the time complexity of MKP algorithms. The state-of-the-art has reduced the complexity from a trivial $ O^*(2^n) $ to $ O^*(c_k^n) $, with $ c_k > 1.94 $ for $ k \geq 3 $, where $ n $ denotes the vertex number. This paper investigates the MKP using two quantum models: gate-based model and annealing-based model. Two gate-based algorithms, qTKP and qMKP, are proposed to achieve $ O^*(1.42^n) $ time complexity. qTKP integrates quantum search with graph encoding, degree counting, degree comparison, and size determination to find a $ k $-plex of a given size; qMKP uses binary search to progressively identify the maximum solution. Furthermore, by reformulating MKP as a quadratic unconstrained binary optimization problem, we propose qaMKP, the first annealing-based approximation algorithm, which utilizes qubit resources more efficiently than gate-based algorithms. To validate the practical performance, proof-of-principle experiments were conducted using the latest IBM gate-based quantum simulator and D-Wave adiabatic quantum computer. This work holds potential to be applied to a wide range of clique relaxations, e.g., $ n $-clan and $ n $-club.