In　this　paper,　an　adjustable　Q-learning　scheme　is　developed　to　solve　the　discrete-time　nonlinear　zero-sum　game　problem,　which　can　accelerate　the　convergence　rate　of　the　iterative　Q-function　sequence.　First,　the　monotonicity　and　convergence　of　the　iterative　Q-function　sequence　are　analyzed　under　some　conditions.　Moreover,　by　employing　neural　networks,　the　model-free　tracking　control　problem　can　be　overcome　for　zero-sum　games.　Second,　two　practical　algorithms　are　designed　to　guarantee　the　convergence　with　accelerated　learning.　In　one　algorithm,　an　adjustable　acceleration　phase　is　added　to　the　iteration　process　of　Q-learning,　which　can　be　adaptively　terminated　with　convergence　guarantee.　In　another　algorithm,　a　novel　acceleration　function　is　developed,　which　can　adjust　the　relaxation　factor　to　ensure　the　convergence.　Finally,　through　a　simulation　example　with　the　practical　physical　background,　the　fantastic　performance　of　the　developed　algorithm　is　demonstrated　with　neural　networks.　©　2024　Elsevier　Ltd