Can a 2010 iMac function as a Monitor for M4 Mac mini?

+1 no, your new MM needs its own dedicated external display. Target Display Mode has been dead for over 10 years. Please read Target Display Mode and you will see where it says:

"24-inch and 27-inch iMac models introduced in 2009 and 2010

To use any of these iMac models as an external display:

• The iMac used as a display must have macOS High Sierra or earlier installed.
• The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed."

No. Even if Apple had not added restrictions on the video source, your tag line says that your 2010 iMac was a 21.5” one. The 21.5” 2010 iMacs never supported Target Display Mode in the first place.

No: Using an iMac as a monitor - Apple Community

No.

You need to connect an external Display or HDMI TV to the Mac mini.

see > Connect a display to Mac mini - Apple Support

No you can't.