A bi-Hamiltonian structure and separation variables are found for a pair of integrable systems on the Poisson manifold with quadratic Hamiltonians and second integrals of motion that are third- and fourth-degree polynomials. The compatible Poisson bivectors are constructed on Poisson manifold in the class of quadratic Poisson tensors and the separation of variables for the corresponding bi-integrable systems possessing higher degree integrals of motion is discussed. The Poisson bivectors are degenerate and need a special reduction to integrate the equations of motions. It is more convenient to use the coordinates to obtain a more compact description of the corresponding integrable systems. Quadratic linear integrals of motion are found in bi-involution with respect to the Poisson bracket associated with the bivectors and with the quadratic linear tensor.