Perhaps new with macOS 12.1 (my first install of Monterey), the Hot Corner for Quick Notes is initially configured to require holding the ⌘ key when you move the cursor to the designated corner.
Alternatively, switching Quick Note to the lower left Hot Corner addressed the multiple monitor issue for me.
Then I noticed the Hot Corners popup window specifies Active Screen Corners. So I set it back to the lower right corner (without the ⌘). And now it works in that corner on both screens - whether the active one or not (also new with 12.1?).