We propose and analyze numerical schemes for the Q-tensor gradient flow coupled with quasi-entropy. The quasi-entropy is a strictly convex, rotationally invariant elementary function that impose physical constraints. The Q-tensor model, containing a fourth-order term of concentration in the energy functional, could be utilized to describe smectic liquid crystals in confinements. For the gradient flow, in addition to the traditional Dirichlet boundary conditions, we propose a time evolution equation for the spatial outer normal derivative of concentration on the boundary. The numerical scheme is validated to satisfy physical constraints and energy dissipation. Furthermore, we derive the error estimates of first-order in time and two-order in space, and carry out some numerical examples to verify the results.