Quantum computing is an emerging and promising alternative to classical computing due to its ability to perform rapidly complex computations in a parallel manner. In this thesis, we aim to design an audio classification algorithm using a hybrid quantum-classical neural network. The thesis concentrated on healthcare applications and focused specifically on COVID-19 cough sound classification. All machine learning algorithms developed or implemented in this study were trained using features from Log Mel Spectrograms of healthy and COVID-19 coughing audio. Results are first presented from a study in which an ensemble of a VGG13, CRNN, GCNN, and GCRNN are utilized to classify audio using classical computing. Then, improved results attained using an optimized VGG13 neural network are presented. Finally, our quantum-classical hybrid neural network is designed and assessed in terms of accuracy and number of quantum layers and qubits. Comparisons are made to classical recurrent and convolutional neural networks.