Accurate detection and segmentation of intracranial aneurysms (IAs) from time-of-flight magnetic resonance angiography (TOF-MRA) scans is critical for effective clinical management and rupture risk assessment. However, IAs are small, sparsely distributed structures with subtle morphological features, making their identification challenging, particularly when annotated datasets are limited or weakly labeled. In this work, we propose a novel quantum-enhanced multi-task network architecture that combines 3D UNet-based feature extraction, soft vesselness priors, and quantum kernel embeddings at the bottleneck to jointly perform voxel-wise aneurysm segmentation and patch-wise aneurysm detection. The shared encoder integrates both the raw MRA image and corresponding vesselness maps, enabling parameter-efficient learning while guiding attention toward vascular structures. At the bottleneck, a quantum kernel layer maps multi-scale features into a high-dimensional Hilbert space, enhancing the representational power and capturing complex correlations between geometric and intensity-based features. Task-specific decoders employ attention mechanisms and multi-scale fusion to produce accurate segmentation masks and classification logits. We evaluate our model on two publicly available TOF-MRA datasets: Lausanne and ADAM, using standard detection metrics (false positive rate, sensitivity) and segmentation metrics (Dice, IoU, 95%-Hausdorff distance). Our quantum-enhanced architecture achieves state-of-the-art performance, reducing false positives while maintaining high sensitivity, and demonstrates robust generalization on external data. Extensive ablation studies highlight the contributions of the vesselness prior, attention gating, test-time augmentation, and quantum kernel integration to overall performance. This work demonstrates that quantum-inspired feature embeddings can significantly improve weakly supervised IA analysis and pave the way for more reliable, automated clinical tools for aneurysm diagnosis and monitoring.