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The increasing penetration of photovoltaic (PV) generation introduces significant uncertainty into power system operation, necessitating forecasting approaches that extend beyond deterministic point predictions. This paper proposes an any-quantile probabilistic forecasting framework for multi-regional PV power generation based on the Any-Quantile Recurrent Neural Network (AQ-RNN). The model integrates an any-quantile forecasting paradigm with a dual-track recurrent architecture that jointly processes series-specific and cross-regional contextual information, supported by dilated recurrent cells, patch-based temporal modeling, and a dynamic ensemble mechanism. The proposed framework enables the estimation of calibrated conditional quantiles at arbitrary probability levels within a single trained model and effectively exploits spatial dependencies to enhance robustness at the system level. The approach is evaluated using 30 years of hourly PV generation data from 259 European regions and compared against established statistical and neural probabilistic baselines. The results demonstrate consistent improvements in forecast accuracy, calibration, and prediction interval quality, underscoring the suitability of the proposed method for uncertainty-aware energy management and operational decision-making in renewable-dominated power systems.