So apparently they mean links to search console itself, so like if you have a link on a blog post to search console or if you have an app that links to search console for some reason, that's what they're talking about.
I was confused too. They wrote it really cryptically, as though I needed to update some link I didn't know about on my own website that somehow links my site to search console, but apparently they just wrote the alert poorly.
Here's some more information on it by people who know more than I do. I asked the same question there and got a response from someone who knows what's up.