Statement-I: Brown ring test can be done for ${\mathrm{NO}}_3^{-}$ in presence of ${\mathrm{NO}}_2^{-}$.

Statement-2: Oxidation state of iron is changing from +2 to + 1 in the brown ring complex.

Statement-1 is False, Statement-2 is True.

NO has an oxidation state of +1in the brown ring complex.

Let oxidation state of Fe be x. 

Given that $\left[\mathrm{Fe}{\left({\mathrm{H}}_2\mathrm{O}\right)}_5{\mathrm{NO}}^{+}\right]$ has a +2 overall charge on the coordination sphere, the total amount of oxidation. 

All of its constituent parts should have states of +2.

Therefore,
$\begin{array}{l}x+1=+2\\ x=+1\end{array}$

Thus, oxidation state of iron is changing from +2 to + 1 in the brown ring complex. Hence, option D is correct.