You would not use "quarto" to say, "The fourth person to arrive", would you? This quarto is only referring to the fraction 1/4, so the "proper" translation should be "quarter". Non è forse vero?
It refers to both: ordinals in Italian are primo (1st), secondo (2nd), terzo (3rd), quarto (4th), quinto (5th), sesto (6th), settimo (7th), ottavo (8th), nono (9th), decimo (10th), undicesimo (11th) and so on. They work as adjectives so they vary with gender and number. As for fractions, 1/4 is "un quarto", 3/4 "tre quarti".