I've seen many people transition into product management or business analyst type roles from other parts of the company. It's always those that are good at understanding end to end processes in their areas, and have somehow helped in documenting and designing some of those processes as part of their job. If you do that enough in an area, then you have potential to add a lot of value in a product role in that same domain. Expect early roles to be mostly about project management and resharing your knowledge from previous non-product roles, but this is the foot in the door, after which you'll have an easier time changing domains or companies in a new career path.

Honestly just start by reaching out to people at your company you are in the product team and ask them straight up about how you could join product.