Is Daniel of the lions den famethe son of king david?

Yes, King David was a prophet according to both the Old and New Testament.

The prophets who wrote down their prophecies (think of the Old Testament Major and Minor Prophets like Isaiah, Jeremiah, etc.) declared their prophecy by saying God (YHWH) was speaking directly to them.

David says this in 2 Samuel 23.

The book of Acts in the New Testament also declares King David a prophet in Acts 2:29-35.