We consider the problem of distributed representation of signals in sensor networks, where sensors exchange quantized information with their neighbors. The signals of interest are assumed to have a sparse representation with spectral graph dictionaries. We further model the spectral dictionaries as polynomials of the graph Laplacian operator. We first study the impact of the quantization noise in the distributed computation of matrix-vector multiplications, such as the forward and the adjoint operator, which are used in many classical signal processing tasks. It occurs that the performance is clearly penalized by the quantization noise, whose impact directly depends on the structure of the spectral graph dictionary. Next, we focus on the problem of sparse signal representation and propose an algorithm to learn polynomial graph dictionaries that are both adapted to the graph signals of interest and robust to quantization noise. Simulation results show that the learned dictionaries are efficient in processing graph signals in sensor networks where bandwidth constraints impose quantization of the messages exchanged in the network.