Since the COVID-19 outbreak, a major scientific effort has been made by researchers and companies worldwide to develop a digital diagnostic tool to screen this disease through some biomedical signals, such as cough, and speech. Joint time–frequency feature extraction techniques and machine learning (ML)-based models have been widely explored in respiratory diseases such as influenza, pertussis, and COVID-19 to find biomarkers from human respiratory system-generated acoustic sounds. In recent years, a variety of techniques for discriminating textures and computationally efficient local texture descriptors have been introduced, such as local binary patterns and local ternary patterns, among others. In this work, we propose an audio texture analysis of sounds emitted by subjects in suspicion of COVID-19 infection using time–frequency spectrograms. This approach of the feature extraction method has not been widely used for biomedical sounds, particularly for COVID-19 or respiratory diseases. We hypothesize that this textural sound analysis based on local binary patterns and local ternary patterns enables us to obtain a better classification model by discriminating both people with COVID-19 and healthy subjects. Cough, speech, and breath sounds from the INTERSPEECH 2021 ComParE and Cambridge KDD databases have been processed and analyzed to evaluate our proposed feature extraction method with ML techniques in order to distinguish between positive or negative for COVID-19 sounds. The results have been evaluated in terms of an unweighted average recall (UAR). The results show that the proposed method has performed well for cough, speech, and breath sound classification, with a UAR up to 100.00%, 60.67%, and 95.00%, respectively, to infer COVID-19 infection, which serves as an effective tool to perform a preliminary screening of COVID-19.