Deploying deep learning models in safety-critical applications remains a very challenging task, mandating the provision of assurances for the dependable operation of these models. Uncertainty quantification (UQ) methods estimate the model's confidence per prediction, informing decision-making by considering the effect of randomness and model misspecification. Despite the advances of state-of-the-art UQ methods, they are computationally expensive or produce conservative prediction sets/intervals. We introduce MC-CP, a novel hybrid UQ method that combines a new adaptive Monte Carlo (MC) dropout method with conformal prediction (CP). MC-CP adaptively modulates the traditional MC dropout at runtime to save memory and computation resources, enabling predictions to be consumed by CP, yielding robust prediction sets/intervals. Throughout comprehensive experiments, we show that MC-CP delivers significant improvements over advanced UQ methods, like MC dropout, RAPS and CQR, both in classification and regression benchmarks. MC-CP can be easily added to existing models, making its deployment simple.