You can but it may be difficult as the battery - part #1-756-710-12 -supplier example only is soldered onto a board. The battery example above was only to show what it looked like and the price of the part. There are other suppliers that may suit you better. Just search online using the part number only in the search term.
Here’s a link to the service manual for the camera. Scroll to p.2-6 and find part BT001 to see the location of the battery.
Here’s a link to the ifixit Sony Cyber-shot DSC-HX20V repair guides, which may be of some help"