<p>This paper is devoted to the study of a class of two-stage adjustable robust quadratic optimization problems with affine decision rules, where both the objective and constraint functions involve spectrahedral uncertain data. By using a new robust constraint qualification condition, necessary and sufficient conditions expressed in terms of linear matrix inequalities are established for the optimal solutions of the adjustable robust quadratic optimization problem. Subsequently, based on the obtained optimality conditions, a semidefinite linear programming (SDP) dual problem of this adjustable robust quadratic optimization problem is proposed. Then, weak and strong duality properties between them are established. The obtained results provide us with a way to find an optimal value of a two-stage adjustable robust quadratic optimization problem by solving its SDP linear dual problem. Furthermore, as a special case, the second-order cone programming (SOCP) dual for the two-stage adjustable robust separable quadratic optimization is considered.</p>