The short answer is: no. The "medium length" answer is: It's not possible, but you can achieve this using a virtual layer.

The long answer is: I is not to be possible to create the coordinates using expressions. Geometry generator does not create individual points (vertices). So you can't create new features, every vertex of the polygon will always be part of the polygon feature, so you can't address the points individually to get their coordinates. Also the attribute table of the polygons has just one entry per polygon feature, so how could you individualise the points there? "it is not possible" is not a solution, but at least an answer.

But there is a good alternative: You can create a virtual layer with this solution proposed by @J. Monticolo: https://gis.stackexchange.com/a/382382/88814 - it works great even for complex multipart geometries with holes.

You can than very easily create a (virtual) field on this virtual layer for the coordinates using $x and $y.

If you just need a list of coordinates, the answer by @Dror Bogin is the way to got.

If you want to have coordinates per vertex (e.g. to label them), you can't do this using without creating a new layer. You can create a virtual layer with this solution proposed by @J. Monticolo: https://gis.stackexchange.com/a/382382/88814 - it works great even for complex multipart geometries with holes.

You can than very easily create a (virtual) field on this virtual layer for the coordinates using $x and $y.

Geometry generator however does not create individual points (vertices). So you can't create new features, every vertex of the polygon will always be part of the polygon feature, so you can't address the points individually to get their coordinates. Also the attribute table of the polygons has just one entry per polygon feature, so you can't individualise the points there.