Video-language models (VLMs) are reshaping video querying services, bringing unified solutions to complex perception and reasoning tasks. However, deploying large VLMs in real-world systems remains challenging due to their high resource demands, and remote-based deployment often results in unacceptable response delays. Although small, locally deployable VLMs offer faster responses, they unavoidably fall short in accuracy. To reconcile this trade-off, we propose QuickGrasp, a responsive, quality of service (QoS)-aware system that bridges this gap through a local-first architecture with on-demand edge augmentation. Built upon the highly modular architecture of VLMs, QuickGrasp shares the vision representation across model variants to avoid redundant computation. To maximize system-wide efficiency, QuickGrasp introduces three key designs: accelerated video tokenization, query-adaptive edge augmentation, and delay-aware, accuracy-preserving vision token density configuration. We implement a prototype of QuickGrasp and evaluate it across multiple video understanding benchmarks. The results show that QuickGrasp matches the accuracy of large VLMs while achieving up to a 12.8x reduction in response delay. QuickGrasp represents a key advancement toward building responsive video querying services for open-world understanding that fully leverage the capabilities of VLMs.