This article explores the discrete-time stochastic optimal LQR control with delay and quadratic constraints. The inclusion of delay, compared to delay-free optimal LQR control with quadratic constraints, significantly increases the complexity of the problem. Using Lagrangian duality, the optimal control is obtained by solving the Riccati-ZXL equation in conjunction with a gradient ascent algorithm. Specifically, the parameterized optimal controller and cost function are derived by solving the Riccati-ZXL equation, with a gradient ascent algorithm determining the optimal parameter. The primary contribution of this work is presenting the optimal control as a feedback mechanism based on the state's conditional expectation, wherein the gain is determined using the Riccati-ZXL equation and the gradient ascent algorithm. Numerical examples demonstrate the effectiveness of the obtained results.