Consider a language $ L_1=\left \{ a^{n} b^{n} \mid n\geq 0 \right \}$ is a non regular language .
And take another language $ L_2={L_1}^{c}$ which is also non regular .(Since regular set is closed under complementation.)
As we know the union of any language and its complement is $\Sigma^*$.
So, $L_1\cup L_2=\Sigma^*$ and this is regular.