Federated machine learning (FL) is a powerful technology which can be implemented to exploit the sheer amount of geographically distributed data for enhanced computation. Exploiting the impending proliferation of wireless devices, in this paper, we incorporate distributed quantized transmissions for reliable connectivity to a remote FL server. We develop a novel theoretical framework for the convergence analysis of the proposed network under joint impact of communication bit error rate (BER), and model quantization, and participation control. We show that the convergence rate of the network is affected by the BER and it can be improved via participation control. Through simulation, we demonstrate that our proposed model can provide the same performance as the conventional FL networks based on point-to-point communication while the energy consumption is divided across the distributed nodes.