This　paper　presents　a　Newton-CG　augmented　Lagrangian　method　for　solving　convex　quadratically　constrained　quadratic　semidefinite　programming　(QCQSDP)　problems.　Based　on　the　Robinson＇s　CQ,　the　strong　second　order　sufficient　condition,　and　the　constraint　nondegeneracy　conditions,　we　analyze　the　global　convergence　of　the　proposed　method.　For　the　inner　problems,　we　prove　the　equivalence　between　the　positive　definiteness　of　the　generalized　Hessian　of　the　objective　functions　in　those　inner　problems　and　the　constraint　nondegeneracy　of　the　corresponding　dual　problems,　which　guarantees　the　superlinear　convergence　of　the　inexact　semismooth　Newton-CG　method　to　solve　the　inner　problem.　Numerical　experiments　show　that　the　proposed　method　is　very　efficient　to　solve　the　large-scale　convex　QCQSDP　problems.