This paper presents a novel Fourier spectral method that utilizes optimization techniques to ensure the positivity and conservation of moments in the space of trigonometric polynomials. We rigorously analyze the accuracy of the new method and prove that it maintains spectral accuracy. To solve the optimization problem, we propose an efficient Newton solver that has a quadratic convergence rate. Numerical examples are provided to demonstrate the high accuracy of the proposed method. Our method is also integrated into the spectral solver of the Boltzmann equation, showing the benefit of our approach in applications.