We study a problem of birational equivalence for polynomial Poisson algebras over a field of arbitrary characteristic. More precisely, the quadratic Gel'fand-Kirillov problem asks whether the field of fractions of a given polynomial Poisson algebra is isomorphic (as Poisson algebra) to a Poisson affine field, that is the field of fractions of a polynomial algebra (in several variables) where the Poisson bracket of two generators is equal to their product (up to a scalar). We answer positively this question for a large class of polynomial Poisson algebras and their Poisson prime quotients. For instance, this class includes Poisson determinantals varieties.