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In vision-language models (VLMs), misalignment between textual descriptions and visual coordinates often induces hallucinations. This issue becomes particularly severe in dense prediction tasks such as spatial-temporal video grounding (STVG). Prior approaches typically focus on enhancing visual-textual alignment or attaching auxiliary decoders. However, these strategies inevitably introduce additional trainable modules, leading to significant annotation costs and computational overhead. In this work, we propose a novel visual prompting paradigm that avoids the difficult problem of aligning coordinates across modalities. Specifically, we reformulate per-frame coordinate prediction as a compact instance-level identification problem by assigning each object a unique, temporally consistent ID. These IDs are embedded into the video as visual prompts, providing explicit and interpretable inputs to the VLMs. Furthermore, we introduce STVG-R1, the first reinforcement learning framework for STVG, which employs a task-driven reward to jointly optimize temporal accuracy, spatial consistency, and structural format regularization. Extensive experiments on six benchmarks demonstrate the effectiveness of our approach. STVG-R1 surpasses the baseline Qwen2.5-VL-7B by a remarkable margin of 20.9% on m_IoU on the HCSTVG-v2 benchmark, establishing a new state of the art (SOTA). Surprisingly, STVG-R1 also exhibits strong zero-shot generalization to multi-object referring video object segmentation tasks, achieving a SOTA 47.3% J&F on MeViS.