No. You can have dragons without that.
In some hidden places around the world you can meet female dragons (not princess). Sometimes, you can bang them and they leave eggs.
You can also buy a dragon egg once during one of the other princess story line (during the festival...).
Yeah I found a dragon egg in a playthrough but the ability to fly with her was unlocked only after wyvern princess joined, atleast that's how I remember it, could be wrong though.