Generative query suggestion aims to enhance user engagement by anticipating user intents and recommending relevant follow-up queries. A central challenge is to generate slates whose individual queries are useful while the slate covers distinct intents. We propose an Intent-Driven Query Suggestion Framework with dual-stage optimization. First, intent-aware diversity modeling constructs intent-aligned supervised fine-tuning (SFT) data and uses an Intent-Aware Diversity Reward to optimize intent coverage. Second, query-level credit assignment routes individual quality signals to the corresponding query tokens while sharing a slate-level diversity signal across the slate. Experiments on a large-scale production dataset, including online A/B testing and offline evaluation, show improvements in click-through rate, query quality, and intent coverage.