This　paper　proposes　an　adaptive　quasi-maximum　likelihood　estimation　(QMLE)　when　forecasting　the　volatility　of　financial　data　with　the　generalized　autoregressive　conditional　heteroscedasticity　(GARCH)　model.　When　the　distribution　of　volatility　data　is　unspecified　or　heavy-tailed,　we　worked　out　adaptive　QMLE　based　on　data　by　using　the　scale　parameter　(f)　to　identify　the　discrepancy　between　wrongly　specified　innovation　density　and　the　true　innovation　density.　With　only　a　few　assumptions,　this　adaptive　approach　is　consistent　and　asymptotically　normal.　Moreover,　it　gains　better　efficiency　under　the　condition　that　innovation　error　is　heavy-tailed.　Finally,　simulation　studies　and　an　application　show　its　advantage.