In this paper, we develop an effective conservative high order finite difference scheme with a Fourier spectral method for solving the inviscid surface quasi-geostrophic equations, which include a spectral fractional Laplacian determining the vorticity for the transport velocity of the potential temperature. The fractional Laplacian is approximated by a Fourier-Galerkin spectral method, while the time evolution of the potential temperature is discretized by a high order conservative finite difference scheme. Weighted essentially non-oscillatory (WENO) reconstructions are also considered for comparison. Due to a low regularity of problems involving such a fractional Laplacian, especially in the critical or supercritical regime, directly applying the Fourier spectral method leads to a very oscillatory transport velocity associated with the gradient of the vorticity, e.g. around smooth extrema. Instead of using an artificial filter, we propose to reconstruct the velocity from the vorticity with central difference discretizations. Numerical results are performed to demonstrate the good performance of our proposed approach.