This paper presents a novel federated learning solution, QHetFed, suitable for Internet of Things deployments, addressing the challenges of clustered geographic distribution, communication resource limitation, and data heterogeneity. QHetFed is based on hierarchical federated learning over multiple device clusters, where the learning process and learning parameters take the necessary data quantization and the data heterogeneity into consideration to achieve high accuracy and fast convergence. Unlike conventional hierarchical federated learning algorithms, the proposed approach combines gradient aggregation in intra-cluster iterations with model aggregation in inter-cluster iterations. We offer a comprehensive analytical framework to evaluate its optimality gap and convergence rate, and give a closed form expression for the optimal learning parameters under a deadline, that accounts for communication and computation times. Our findings reveal that QHetFed consistently achieves high learning accuracy and significantly outperforms other hierarchical algorithms, particularly under heterogeneous data distributions.