from the thread at https://skyline.ms/announcements/home/support/thread.view?rowId=51690

which culminates in this tip:

"Brendan MacLean (Members, UGM Registrees) responded    2021-07-16
A little known secret:
Selecting a protein element in the Targets tree, Edit > Copy followed by a paste into a text editor such as Notepad gives you FASTA for the protein. And this works for multiple selections.

So, I would:
1. Reduce the document to only the proteins you want in your FASTA.
2. Edit > Collapse All > Proteins
3. Edit > Select All
4. Edit > Copy
5. Paste into a text editor.

We could make this less well hidden and easier to use, but this should get you what you are looking for.

Hope this helps. Thanks for posting to the Skyline support board.

—Brendan"

Wrapping this functionality in a File>Export>FASTA menu item would be pretty easy to do, and would make this easier to find.