Distributed graph signal processing methods require that the graph nodes communicate by exchanging messages. These messages have a finite precision in a realistic network, which may necessitate to implement quantization. Quantization, in turn, generates errors in the distributed processing tasks, com- pared to perfect settings. This paper proposes a novel method to minimize the quantization error without compromising the communication costs by bounding the exchanged messages along with allocating a limited bit budget through the network in an optimized way. In particular, the quantization adapts to the network topology and message importance in the iterative distributed processing algorithm. Our results show that the proposed method is efficient in minimizing the quantization error and that it outperforms baseline algorithms when the bit budget is limited.