By using the brick-wall model, the quantum entropy of a cylindrical black hole in Anti-de Sitter space-time arising from the massless Dirac field is discussed. It is shown that the quantum entropy has both linear and logarithmical divergences if the usual contribution from the vacuum surrounding the system is ignored, and the whole expression does not take the form of the scalar one. Both the whole logarithmic term and the subleading logarithmic term which relates to the spins of the Dirac particles are always positive.