The rapid growth of large language model (LLM) deployments has made cost-efficient serving systems essential. Recent efforts to enhance system cost-efficiency adopt two main perspectives: (i) An algorithmic perspective that exploits heterogeneous model capabilities to route simpler queries to lower-cost models and complex queries to higher-cost models (i.e., heterogeneous query routing); and (ii) a systems perspective that utilizes heterogeneous GPU resources as cost-effective alternatives to homogeneous high-end GPUs (i.e., heterogeneous model deployment). However, algorithm-system co-design for cost-efficient LLM serving necessitates sophisticated management: (i) Determining optimal query routing strategies under latency and quality requirements, (ii) configuring model deployment across heterogeneous GPUs with appropriate resource allocation and parallelism strategies, and (iii) co-optimizing routing and deployment decisions to maximize overall system performance. To address these challenges, we present BOute, a quality-aware scheduling system that jointly exploits heterogeneous model and GPU capabilities for cost-efficient LLM serving. BOute employs a multi-objective Bayesian optimization (MOBO) framework to co-optimize the routing strategy and model deployment, thereby maximizing the cost-efficiency of the serving system while guaranteeing response quality. Evaluation results demonstrate that BOute outperforms state-of-the-art LLM serving systems by up to 157% and 59% on average under identical cost budgets and quality requirements, or reducing serving costs by 15%-61% (38% on average) while maintaining the same performance targets, validating its effectiveness in achieving cost-efficient LLM serving.