We address the product question generation task. For a given product description, our goal is to generate questions that reflect potential user information needs that are either missing or not well covered in the description. Moreover, we wish to cover diverse user information needs that may span a multitude of product types. To this end, we first show how the T5 pre-trained Transformer encoder-decoder model can be fine-tuned for the task. Yet, while the T5 generated questions have a reasonable quality compared to the state-of-the-art method for the task (KPCNet), many of such questions are still too general, resulting in a sub-optimal global question diversity. As an alternative, we propose a novel learning-to-diversify (LTD) fine-tuning approach that allows to enrich the language learned by the underlying Transformer model. Our empirical evaluation shows that, using our approach significantly improves the global diversity of the underlying Transformer model, while preserves, as much as possible, its generation relevance.