see the screenshot above. the page entity does not expose fields, sorting, grouping, field names etc. (ie its not possible to see those things when you open the page entity in shotgun).
there is also no field that contains a JSON representation of the page details.
so based on what i see in shotgun its not possible but I’m hopeful someone might have some API ninja tricks that are not easily visible
Try the Fields Admin page, it displays way more fields that are not usually visible in the main UI as they are backend fields.
Some of these things, as I mentioned, are available when you run an AMI from the page.
For example, which fields are visible on the page, what you have selected etc.
Also a users individual page settings (like filters) may be saved on a hidden field somewhere on their user perhaps.
For example, each user has a hidden field that records if they have gone through the first user login tutorial. There may be a lot you can query that is not part of official docs but still queryable.