Models of normal open induction (NOI) are those discretely ordered rings, integrally closed in their fraction field whose nonnegative part satisfy Peano's induction axioms for open formulas in the language of ordered semirings. Here we study the problem of representability of an element $a$ of a model $M$ of NOI (in some extension of $M$) by a quadratic form of the type $X^2 + bY^2$ where $b$ is a nonzero integer. Using either a trigonometric or a hyperbolic parametrization we prove that except in some trivial cases, $M\lbrack x, y\rbrack$ with $x^2 + by^2 = a$ can be embedded in a model of NOI. We also study quadratic extensions of a model $M$ of NOI; we first prove some properties of the ring of Gaussian integers of $M$. Then we study the group of solutions of a Pell equation in NOI; we construct a model in which the quotient group by the squares has size continuum.