A hidden semi-Markov-switching quantile regression model is introduced as an extension of the hidden Markov-switching one. The proposed model allows for arbitrary sojourn-time distributions in the states of the Markov-switching chain. Parameters estimation is carried out via maximum likelihood estimation method using the Asymmetric Laplace distribution. As a by product of the model specification, the formulae and methods for forecasting, the state prediction, decoding and model checking that exist for ordinary hidden Markov-switching models can be applied to the proposed model. A simulation study to investigate the behaviour of the proposed model is performed covering several experimental settings. The empirical analysis studies the relationship between the stock index from the emerging market of China and those from the advanced markets, and investigates the determinants of high levels of pollution in an Italian small city.