I had a new vinyl floor put around 8 months ago. There is a square in the middle where it has damaged the floor. It is in the middle of the kitchen floor. Is there anything I can do to make this look better without having to replace the whole floor?
If, by "a square" you mean that you have rigid vinyl tile flooring (that typically comes in 12"x12" squares) you can pull up and replace a single tile (or have that done, depending on your preference.)
If you have sheet-type vinyl flooring where one piece covers the whole floor, you either need to accept some new seams or replace the whole thing.
Or put a rug over the damaged area ;-)