@Andrei Tulbure no, conformal prediction is a general framework it does not produce quantiles as such what it produces are Prediction Intervals where true values are guarantees to lie in with user specified probability. Classical quantile regression does not produce true quantiles - just because it says 75% quantile there are no guarantees there will be 75 percent of observations below it. You can see an example of what it does and what is the difference here. NeuralProphet trained with quantile regression shown to provide miscalibrated intervals but Conformal Prediction has corrected it. So it can both produce prediction intervals that are well calibrated or alternatively correct prediction intervals from other models.
https://valeman.medium.com/probabilistic-forecasting-with-conformal-prediction-and-neuralprophet-af9c87901d94