Are all the Ultegra brake pads the same? I have a road bike with Shimano Ultegra Di2 6870 goupset. Today it was my first long ride with it and alsmost instantly I noticed that the brake pads are not for carbon rims. I tried searching on the internet brake pads for ultegra 6870 but with no luck. Are other ultegra brake pads compatible with my brakes?
If you have carbon rims (with a carbon braking surface) you will need appropriate brake pads, so as not to damage the rims.  You do not have to use shimano brake shoes- most (possibly all?) rim brakes will take any brand of shoes. 

There is a carbon rim insert for Dura Ace, which might fit the ultegra shoes.  However if there are no compatible pads for your brake shoes/ rims, you may have to buy a new set of shoes - not especially pricy, and they will come with pads.