The usage of smartphone-collected respiratory sound, trained with deep learning models, for detecting and classifying COVID-19 becomes popular recently. It removes the need for in-person testing procedures especially for rural regions where related medical supplies, experienced workers, and equipment are limited. However, existing sound-based diagnostic approaches are trained in a fully supervised manner, which requires large scale well-labelled data. It is critical to discover new methods to leverage unlabelled respiratory data, which can be obtained more easily. In this paper, we propose a novel self-supervised learning enabled framework for COVID-19 cough classification. A contrastive pre-training phase is introduced to train a Transformer-based feature encoder with unlabelled data. Specifically, we design a random masking mechanism to learn robust representations of respiratory sounds. The pre-trained feature encoder is then fine-tuned in the downstream phase to perform cough classification. In addition, different ensembles with varied random masking rates are also explored in the downstream phase. Through extensive evaluations, we demonstrate that the proposed contrastive pre-training, the random masking mechanism, and the ensemble architecture contribute to improving cough classification performance.