By the ramification hypothesis, the reduced norm $\mathrm{nrd}$ gives a positive definite quadratic form $\mathrm{Tr}\circ \mathrm{nrd} : \mathcal{O}\to \mathbb{Q}$. On the other hand, the reduced norm induces a morphism $f : \mathcal{O}^\times/\mathcal{O}_F^\times \to \mathcal{O}_F^\times/(\mathcal{O}_F^\times)^2$. The latter group is finite, and you can compute it by standard algorithms for units in number fields. For each $u\in \mathcal{O}_F^\times$ in a set of representatives modulo squares, the fiber $f^{-1}(u)$ is finite and admits representatives in the finite set $\{x\in\mathcal{O} \mid \mathrm{Tr}(\mathrm{nrd}(x)) = \mathrm{Tr}(u)\}$ which you can compute by lattice enumeration (Fincke-Pohst algorithm).