We focus on a particular class of bilevel programs with a quadratic lower level, which can be obtained reformulating semi-infinite problems with an infinite number of quadratically parametrized constraints. We propose a new approach to solve this class of bilevel programs, based on the dual of the lower level problem and which can lead to a convex or a semidefinite programming problem depending on the parametrization of the lower level with respect to the upper level variables. This approach is compared with is a tailored cutting-plane algorithm, of which we discuss the convergence. We test these two methods on several instances of two applications: the constrained quadratic regression and a zero-sum game with cubic payoff.