There is actually no relationship mentioned throughout the game for Terra and Edgar. No, the relationship is strictly fan based. When Edgar first meets Terra, she is still unsure about herself since she has no memory of anything prior to having the Slave Crown removed. Edgar only knows her as the "witch." However, upon first meeting her, he tries his normal tactics of flirting with her since she is a beautiful young woman. Nothing seems to work, and he at first he thinks he is getting rusty. However, that is not true. Terra doesn't know what flirting is or even what the emotion love is. However, that doesn't break them apart.
Edgar lies to protect her when Kefka stops by his castle looking for her. He even lies again when Kefka stops by and causes fire and destruction to his castle. This causes Edgar, Terra, and Locke to escape his castle upon a daring move leaving his kingdom to submerbed in the desert. This also breaks his hidden allegience with Empire causing future trouble for his kingdom.
At first, Edgar tries to be different than what the Empire wanted. However, Edgar's ideas are similar. He knows Terra's abilities could help solve the war. This thought doesn't escape Terra's mind and causes her to distant herself from the young king. However, throughout the whole game, Edgar isn't trying to use Terra for her powers. He is trying to understand her and earn her trust. Edgar is always worried about Terra. He wants to make sure that she is safe and free. He wants her to understand who she is, but he also wants to see if she could be a solution to the war that everyone is involved in.