A localized differential quadrature method (LDQM) is introduced for buckling analysis of axially functionally graded nonuniform columns with elastic restraints. Weighting coefficients of differential quadrature discretization are obtained making use of neighboring points in forward and backward type schemes for the reference grids near the beginning and end boundaries of the physical domain, respectively, and central type scheme for the reference grids inside the physical domain. Boundary conditions are directly implemented into weighting coefficient matrices, and there is no need to use fictitious points near the boundaries. Compatibility equations are not required because the governing differential equation is discretized only once for each reference grid using neighboring points and variation of flexural rigidity is taken to be continuous in the axial direction. A large case of columns having different variations of cross-sectional profile and modulus of elasticity in the axial direction are considered. The results for nondimensional critical buckling loads are compared to the analytical and numerical results available in the literature. Some new results are also given. Comparison of the results shows the potential of the LDQM for solving such generalized eigenvalue problems governed by fourth-order variable coefficient differential equations with high accuracy and less computational effort.