An infinite-dimensional convex optimization problem with the linear-quadratic cost function and linear-quadratic constraints is considered, We generalize the interior-point techniques of Nesterov-Nemirovsky to this infinite-dimensional situation. The complexity estimates obtained are similar to finite-dimensional ones. We apply our results to the linear-quadratic control problem with quadratic constraints. It is shown that for this problem the Newton step is basically reduced to the standard LQ problem.