Cannot copy directory in MacOS with terminal
When you want copy a directory with command cp you need add option -r or -a.
-a is best because it preserves the structure and attributes of files as indicated in the man command :
man cp

In your case if QT_screen_recs is a directory the corrects commands are :
cp -r /Users/kylefoley/library/containers/com.apple.QuickTimePlayerX/data/library/Autosave\ Information /Volumes/old/QT_screen_recs

or
cp -a /Users/kylefoley/library/containers/com.apple.QuickTimePlayerX/data/library/Autosave\ Information /Volumes/old/QT_screen_recs