When answering this question, you must consider the reactivity series. Mg is much higher than Zn in the reactivity series, therefore it can displace the Zn from the solution - displacement reaction.
It seems odd that you'd react Magnesium nitrate with zinc nitrate. Maybe you meant magnesium with zinc nitrate or vice Versa. Because in your case each compound would remain as it is as both Mg and Zn are happy in a relatively stable compound.
As for magnesium nitrate and magnesium nitrate, you cannot achieve a reaction between two substances that are the same. The same applies for Mg and magnesium nitrate. No displacement happens.
Hope it helps!
