This paper studies geometric and structural properties of quadratic multiobjective optimization problems. Thereby, a multiobjective optimization problem is called convexlike if the upper image, i.e., the image set plus the nonnegative orthant, is a convex set. Moreover, we say that a feasible point is supported in case it is a minimal solution of a weighted sum of the objective functions with nonnegative weights, and we are interested in the question if all efficient, i.e., optimal, solutions of the multiobjective optimization problem under consideration are supported. We investigate conditions under which weighted sum scalarizations admit an optimal solution and how this is related to the existence of positive (semi-)definite convex combinations of the matrices defining the quadratic terms. An equivalent dual characterization of the existence of such convex combinations is derived. In the convexlike unconstrained setting, we show that the absence of weakly efficient solutions is equivalent to the upper image set being the whole space. For the unconstrained biobjective case, we obtain simplified characterizations of the existence of positive (semi-)definite convex combinations. These results clarify conditions under which weighted sum scalarizations are appropriate for solving quadratic multiobjective optimization problems.