Natural silences should never be awkward between a couple who get on well together. If you feel you have to fill every minute together with conversation, then it can mean you are trying to make up for something else that is missing from the relationship.
You should be enjoying time as a couple. Can't you go to a museum and just appreciate the fact that you are doing something together and look at the exhibits without a running commentary? Or walk together and have fun holding hands and liking the fresh air and sights and sounds? It can be quite intimate having touch without words. Try it, and if it still doesn't work, as above, you aren't compatible.