You can set up the Quicklayers palette, and it will remember its place, but it will not re-open automatically when you open Archicad. You will have to open the palette using the menu command each time.
This is true for some of the palettes and is due to a programming difficulty that GS hasn't decided to prioritize - yet. The complaint is old.
If you want to fix this for yourself you can make a custom Toolbar with those commands, and include it in your work environment. It will work the way you want it.
AC4.1-AC26SWE; MacOS13.5.1; MP5,1+MBP16,1