Can I replace my face mount hinges with hidden ones?
Yes it is possible. You would need Face Frame hinges similar to the one shown below. They come in many sizes depending on how much of an overlay your doors will have over the cabinet frame. 



The drilling instructions differ with the type of hinge you will use so you'd want to figure that out first. Also, I would recommend borrowing, renting a drill press because each door will need two 35mm  blind holes drilled into it.  Good luck