This paper investigates the critical buckling behavior of axially functionally graded (FG) material beams with three end support conditions. The FG materials are assumed to have continuously graded based on a power-law function of the volume fractions of the constituents. The governing equation for buckling is derived and solved using the differential quadrature (DQ) method. A comparison between the results obtained from the DQ method and the analytical approach reveals excellent agreement. The effects of various parameters, such as the gradient index and boundary conditions, on the critical buckling load is thoroughly analyzed. The findings highlight the efficiency and accuracy of the DQ method for analyzing functionally graded beams. Moreover, the insights gained from this study can inform the design and optimization of functionally graded structures.