Sixth-order quasi-compact difference (QCD) schemes are proposed for the two-dimensional (2D) and the three-dimensional (3D) Helmholtz equations with the variable parameter. Our approach provides the compact mesh stencil for the unknowns, while the noncompact mesh stencil is employed for the source term and the parameter function without involving their derivatives. For the proper interior grid points that are without adjoining the boundary, the sixth-order truncated errors are obtained by the QCD method. Yet the compact scheme is utilized for both of the source term and the parameter function on the improper interior grids that neighbor the boundary, which only reaches the fourth-order local truncated errors. Theoretically, the sixth-order accuracy of the global error by the proposed QCD method is strictly proved for the non-positive constant parameter. Numerical examples are given to demonstrate that the QCD method achieves the global sixth-order convergence for general variable parameters.