The short answer is you would need to properly install Linux Mint.
You could install all the Mint packages and make Ubuntu look and act like Mint, but it would still be Ubuntu.
Im not saying its impossible to trick the OS to upgrade to Mint, that would require someone more knowledgeable than me, but why risk problems down the road? You have to remember Mint while mint is based on Ubuntu, which is based on Debian, that doesnt mean everything is compatible. Ubuntu might be using a kernel or software package that hasnt been tested in Mint and might not work. Its really not a risk worth taking, considering how easy full installs are.
You can backup your home directory and other data and quickly install Mint and restore your files.