In graph signal processing, signals are processed by explicitly taking into account their underlying structure, which is generally characterized by a graph. In this field, graph filters play a major role to process such signals in the so-called graph frequency domain. In this paper, we focus on the design of autoregressive moving average (ARMA) graph filters and basically present two design approaches. The first approach is inspired by Prony's method, which considers a modified error between the modeled and the desired frequency response. The second approach is based on an iterative method, which finds the filter coefficients by iteratively minimizing the true error (instead of the modified error) between the modeled and the desired frequency response. The performance of the proposed design algorithms is evaluated and compared with finite impulse response (FIR) graph filters. The obtained results show that ARMA filters outperform FIR filters in terms of approximation accuracy even for the same computational cost.