Can I extend my 60-day visa on entry by 30 days at Chamchuri Square immigration?

You can apply for a 30-day tourism extension once at the IT Square Laksi Plaza office if your address is in Bangkok when your entry stamp is 45 days or less remaining. 30 days will be added to your entry stamp.

No, you cannot do it at the Chamchuri Square office as they don't deal with tourist-related applications.

This is general information on tourism extension applications.