Air navigation service providers currently rely on deterministic demand forecasts for Air Traffic Flow Management (ATFM), which inherently fail to quantify forecast uncertainty. This study presents a top-down probabilistic forecasting framework for inbound air traffic demand using Air Traffic Control the Netherlands (LVNL) as a case study. The framework models the uncertainty of an existing deterministic forecasting system using quantile regression, thereby capturing heteroscedasticity directly at the aggregate demand level. To improve empirical coverage under non-exchangeable conditions, Conformalized Quantile Regression is combined with Adaptive Conformal Inference. The framework was applied using approximately two years of operational ATFM data. Compared to a statistical error-margin baseline, the proposed machine learning approach produced substantially narrower prediction intervals and achieved empirical coverage closer to the target, leading to a 25.1% lower Mean Winkler Interval Score (MWIS). Furthermore, Adaptive Conformal Inference more closely tracked the nominal 90% target coverage level over time than static conformal calibration, while also slightly reducing average interval width and MWIS. A retrospective operational analysis indicates that the probabilistic framework enables more flexible risk-based decision-making compared to deterministic forecasting alone. For a representative 150-minute prediction horizon, specific probabilistic decision thresholds simultaneously reduced both the false positive and false negative rates of capacity breaches relative to the deterministic baseline across the evaluated test dataset. These results demonstrate the potential of adaptive conformal prediction techniques to support probabilistic inbound demand forecasting within ATFM operations. ... Read more