Honey bee colony losses threaten global pollination services, yet current monitoring systems treat each hive as an isolated unit, ignoring the spatial pathways through which diseases spread across apiaries. This paper introduces the Spatio-Temporal Apiary Graph Convolutional Network (STAG-CN), a graph neural network that models inter-hive relationships for disease onset prediction. STAG-CN operates on a dual adjacency graph combining physical co-location and climatic sensor correlation among hive sessions, and processes multivariate IoT sensor streams through a temporal--spatial--temporal sandwich architecture built on causal dilated convolutions and Chebyshev spectral graph convolutions. Evaluated on the Korean AI Hub apiculture dataset (dataset \#71488) with expanding-window temporal cross-validation, STAG-CN achieves an F1 score of 0.607 at a three-day forecast horizon. An ablation study reveals that the climatic adjacency matrix alone matches full-model performance (F1\,=\,0.607), while the physical adjacency alone yields F1\,=\,0.274, indicating that shared environmental response patterns carry stronger predictive signal than spatial proximity for disease onset. These results establish a proof-of-concept for graph-based biosecurity monitoring in precision apiculture, demonstrating that inter-hive sensor correlations encode disease-relevant information invisible to single-hive approaches.