What can someone who has an extensive marketing background do to prove themselves competent for an entry-level demand generation role?

That depends on what "extensive marketing background" means.

If you've been, say, a Social Media Marketer and are looking to make a change, I would suggest having a conversation with your boss and the DG leader at your current organization to see if there is any between-teams movement available. There might not be anything right now, but that could change in 6 months or 1 year and then you will be top of mind. In the meantime, they might even be able to have you work on some small projects to learn more about the function and what you can do.

If you've had internships and majored in marketing, but no formal work experience, then I would instead recommend learning about the role through reading forums like this one and talking to people in the role currently. When interviewing candidates for entry-level roles, one of the things I look for most is fit - does the candidate actually want THIS job, not just any job. But showcasing that you understand and really want the role, you will have a leg up on others.