We propose a minimax design method for finite impulse response (FIR) graph filters in this paper. We first convert the graph filter design problem into a numerically stable equivalent problem using a shifted monomial basis. Then, we convert the minimax graph filter design problem into a semidefinite programming problem. Furthermore, we propose a computationally-efficient structure to implement the FIR graph filters designed using the proposed method. Design examples confirm that the proposed implementation structure substantially saves additions required to process a graph signal compared to graph filters designed using a previously proposed Chebyshev polynomial approximation method.