You can always rewrite a subobject $V \subseteq \mathbb{Q}$ as a function $\mathbb{Q} \to \Omega$, but you'll need to includes all the axiom that are in the deginition.
Even if you only look at defining upper Dedekind cut the definition you are proposing is missing the fact that if $x \in V$ then there exists an $x' <x$ such that $x' \in V$.
And upper Dedekind only get you so far:
The more interesting notion of real number (like the Dedekind/continus, or MacNeille) requier a pair of such functions for their definition one encodage $x<q$ and one for $q<x$.
And for most definition, you need actually two subsets