Cough is a frequent symptom in many common respiratory diseases and is considered a predictor of early exacerbation or even disease progression. Continuous cough monitoring offers valuable insights into treatment effectiveness, aiding healthcare providers in timely intervention to prevent exacerbations and hospitalizations. Objective cough monitoring methods have emerged as superior alternatives to subjective methods like questionnaires. In recent years, cough has been monitored using wearable devices equipped with microphones. However, the discrimination of cough sounds from background noise has been shown a particular challenge. This study aimed to demonstrate the effectiveness of single-axis acceleration signals combined with state-of-the-art deep learning (DL) algorithms to distinguish intentional coughing from sounds like speech, laugh, or throat noises. Various DL methods (recurrent, convolutional, and deep convolutional neural networks) combined with one- and two-dimensional time and time–frequency representations, such as the signal envelope, kurtogram, wavelet scalogram, mel, Bark, and the equivalent rectangular bandwidth spectrum (ERB) spectrograms, were employed to identify the most effective approach. The optimal strategy, which involved the SqueezeNet model in conjunction with wavelet scalograms, yielded an accuracy and precision of 92.21% and 95.59%, respectively. The proposed method demonstrated its potential for cough monitoring. Future research will focus on validating the system in spontaneous coughing of subjects with respiratory diseases under natural ambulatory conditions.