This is specifically why you should have been using Apple's Time Machine backup to an external drive, for these painful experiences where you need to restore an otherwise lost document from that backup. Back up your Mac with Time Machine - Apple Support
Although iCloud is not a backup solution, if you have accidentally deleted a file there, Apple provides a 30-day sliding window for file recovery. Recover deleted files on iCloud.com - Apple Support
Because Pages will continue to save files in the last saved folder location until you change that, it is still possible that the file may remain on your Mac, just obscurred in its location. When you launch Pages, visit the File menu > Open Recent sub-menu and see if the file is listed there and can be opened.
If not, then try a Spotlight search, in this case, look for any Pages document that was modified more recently (>) than August. Use the Spotlight 🔍 tool on the right menu bar to enter this query:
kind:pages modified:>08/01/2024
and if you locate the file, press ⌘+R to open a Finder window with that specific file selected.
Do note that Pages documents that have not been manually saved at least once are not auto-saving your content and quitting Pages would irrevocably remove any unsaved work.