Connecting a refrigerator water line - modifying the existing tubing
I would install a Ice maker outlet box in the wall. This makes a nice clean looking install that is flush with the wall and does not interfere with getting the fridge in place.
It is like a washing machine valve box but with just one 1/4 turn valve that will except the stainless steel braided line. The braided line screws onto the valve and then onto the tube in the back of your refrigerator.
There are few different kind of connection types available, Copper sweat, Pex crimp and push to connect.
You will need to transition from your 1/4 copper to whatever type connection the box is that you choose.