That's actually quite a tough question. There are many theories as to how the game started and where it draws its influence from for example Persian As-Nas, French Poque, German Pochen and even, Irish Poca all the way back to a Chinese game of domino cards way back in the first millenium.
So ignoring where it may have taken its influences, most people would agree that the modern game developed during the 19th century in the USA