We introduce a new task called *entity-centric question generation* (ECQG), motivated by real-world applications such as topic-specific learning, assisted reading, and fact-checking. The task aims to generate questions from an entity perspective. To solve ECQG, we propose a coherent PLM-based framework GenCONE with two novel modules: content focusing and question verification. The content focusing module first identifies a focus as “what to ask” to form draft questions, and the question verification module refines the questions afterwards by verifying the answerability. We also construct a large-scale open-domain dataset from SQuAD to support this task. Our extensive experiments demonstrate that GenCONE significantly and consistently outperforms various baselines, and two modules are effective and complementary in generating high-quality questions.