Recent years have seen remarkable progress in speech emotion recognition (SER), thanks to advances in deep learning techniques. However, the limited availability of labeled data remains a significant challenge in the field. Self-supervised learning has recently emerged as a promising solution to address this challenge. In this paper, we propose the vector quantized masked autoencoder for speech (VQ-MAE-S), a self-supervised model that is fine-tuned to recognize emotions from speech signals. The VQ-MAE-S model is based on a masked autoencoder (MAE) that operates in the discrete latent space of a vector quantized variational autoencoder. Experimental results show that the proposed VQ-MAE-S model, pre-trained on the VoxCeleb2 dataset and fine-tuned on emotional speech data, outperforms an MAE working on the raw spectrogram representation and other state-of-the-art methods in SER.