My father never meet or knew her father she was told he was half greek!! But every test on market she Is half Italy is her true dad italin? ?

"My father never meet or knew her father"

does that makes sense to you?

Is the Greek father from Ionian sea???

Ionian islands were of Italian occupation and many Italian invaders married local Greek women 

Some Greek women are married to Italian men in Sicily too 

That or her father is Greek but her mother is Italian.

So a kid born in Italy sails to the other side of the Adriatic and lives, works, learns the language in Greece.  Later, when his girlfriend's family asks where he's from, she says "Greece".  Surely that is not an unusual or hard-to-imagine story.  And if fits perfectly with your facts. 

Greek, Italian, to-ma-toe, to-mah-to.

First, you have to understand that people moved pretty freely around Europe until the 20th century. So it's very possible that the genes are Italian, but the person lived in Greece and adopted Greek customs.