"Not that I heard of". Is this a somewhat shortened form of a full sentence?

This is a common idiomatic turn; there is a similar one, "not that I know of" that is used in a similar context but much more frequently (ngram). Apparently, "not that I've heard of" is not used (ngram).

This turn is used in answer to a question. You can consider it to be the shortening of anything that makes sense. For instance, in the dialogue below there are at least four options you can use instead.

• Att.Gen. Was there any discourse at that time Mr Coote?
  Pomfret Not that I heard of, one word.

• Not any discourse that I heard of, one word.
  Not any that I heard of, one word.
  There was not any discourse that I heard of, one word.
  There was not any that I heard of, one word.

Most often, when you can use "not that I heard of", "not that I know of" can be used in its place.

It's a fragment suggesting a response to an interrogative. The right forms are "not that I've heard of" or "not that I know of".