In this paper, we present a communication-efficient federated learning framework inspired by quantized compressed sensing. The presented framework consists of gradient compression for wireless devices and gradient reconstruction for a parameter server (PS). Our strategy for gradient compression is to sequentially perform block sparsification, dimensional reduction, and quantization. Thanks to gradient sparsification and quantization, our strategy can achieve a higher compression ratio than one-bit gradient compression. For accurate aggregation of the local gradients from the compressed signals at the PS, we put forth an approximate minimum mean square error (MMSE) approach for gradient reconstruction using the expectation-maximization generalized-approximate-message-passing (EM-GAMP) algorithm. Assuming Bernoulli Gaussian-mixture prior, this algorithm iteratively updates the posterior mean and variance of local gradients from the compressed signals. We also present a low-complexity approach for the gradient reconstruction. In this approach, we use the Bussgang theorem to aggregate local gradients from the compressed signals, then compute an approximate MMSE estimate of the aggregated gradient using the EM-GAMP algorithm. We also provide a convergence rate analysis of the presented framework. Using the MNIST dataset, we demonstrate that the presented framework achieves almost identical performance with the case that performs no compression, while significantly reducing communication overhead for federated learning.