Is there a free way to use 2013 iMac as a monitor for an M4 Mac Mini?

+1, your iMac cannot be used as an external display for a MM M4. You need to buy an external display. They are inexpensive starting at <$100 and are:

• Simple to setup
• No maintenance necessary
• Extremely reliable.

Nope' the best bet, is to just get and use an external display or two with the M4 Mac mini.

see > Connect a display to Mac mini - Apple Support