Quantum generative models based on IQP circuits hold promise for learning complex distributions, but they suffer from two major limitations: lack of output control and generation bias. To address these issues, this paper presents ConQuER, a controllable quantum generative framework with a modular circuit architecture. ConQuER uses lightweight controller circuits, combined with pre-trained IQP circuits, to precisely control the output distribution without requiring full retraining. Furthermore, data-driven optimization is used to incorporate an intrinsic control path into the IQP architecture, thereby reducing generation bias. ConQuER maintains efficient classical training properties and high scalability, and has been experimentally validated on multiple quantum state datasets.