divination - Could there been some "Seer" Blood in Harry Potter?

The heat was overpowering and his nostrils were stinging with the perfumed smoke wafting from the fire beside them. He thought of what Ron had just said, and decided to pretend.

"Er --" said Harry, "a dark shape... um..."

"What does it resemble?" whispered Professor Trelawney. "Think, now..."

Harry cast his mind around and it landed on Buckbeak.

"A hippogriff," he said firmly.

"Indeed!" whispered Professor Trelawney, scribbling keenly on the parchment perched upon her knees. "My boy, you may well be seeing the outcome of poor Hagrid's trouble with the Ministry of Magic! Look closer... Does the hippogriff appear to... have its head?"

"Yes," said Harry firmly.

"Are you sure?" Professor Trelawney urged him. "Are you quite sure, dear? You don't see it writhing on the ground, perhaps, and a shadowy figure raising an axe behind it?"

"No!" said Harry, starting to feel slightly sick.

"No blood? No weeping Hagrid?"

"No!" said Harry again, wanting more than ever to leave the room and the heat. "It looks fine, it's - - flying away..."

I know some may say that because he didn't want Buckbeak to be killed, or he was pretending. But for pretending, was it a coincidence that his mind landed on Buckbeak. Was it a coincidence that he told her that it was flying away(knowing that he may not have been able to concentrate on what he was saying because the uncomfortable surroundings)? Or was it some seer thingy coming into play that made Harry predict the future?

Answer

Yes, it's possible

The Pottermore article about Sybill Trelawney shows that seer blood can get diluted though the generations until it is almost gone:

Sybill is the great-great granddaughter of a genuine Seer, Cassandra Trelawney. Cassandra's gift has been much diluted over ensuing generations, although Sybill has inherited more than she knows.

Also, as Sirius says, all Wizards are related:

“You’re related to the Malfoys!”
“The pure-blood families are all interrelated,” said Sirius. “If you’re only going to let your sons and daughters marry purebloods your choice is very limited, there are hardly any of us left.
(OotP6)

As Megha points out, the fact that a lot of divination works around him throughout the books (see this question), and that he sees and visions in third person may be indication that he does have seer blood.