I think the following construction could work: Let $X$ be a Calabi-Yau threefold without any rational curves (such threefolds do exist). Then $Nef(X)=\overline{Eff}(X)$, i.e., the nef cone equals the pseudoeffecitve cone (by the log cone theorem) and the nef boundary is given by the null cone, that is, the set of divisor classes $D$ such that $D^3=0$. Note that this null cone is given by the zero locus of a degree 3 polynomial in the Neron Severi group, and the non-rational points of this will give you the example.

I should mention that the nef cone of a Calabi-Yau threefold is a very interesting object even though it is often non-rational polyhedral. Indeed, the Kawamata-Morrison cone conjecture states that the nef cone and movable cone should instead have a rational polyhedral *fundamental domain* for the action of $im(Aut(X)\to GL(N^1(X)))$. In some sense, this is the next best thing compared to the Fano case ($K<0$) where everything is rational polyhedral. So in particular if the automorphism group is infinite, then one would expect a non-rational polyhedral nef cone.