What is the molecular formula of the nitrate of a metal Q that forms a carbonate of the form QCO3?
The carbonate anion is $\ce{CO3^{2-}}$. This tells us that the metal cation is doubly positively charged:
$$ \ce{Q^{2+} + CO3^{2-} -&gt; QCO3} $$

The other two anions mentioned in your question are


the nitrite anion $\ce{NO2^-}$
the nitrate anion $\ce{NO3^-}$


Since the question you're concerned with asks for the formula of the nitrate of $\ce{Q}$, the correct answer would be $\ce{Q(NO3)2}$. 

However, the compound $\ce{Q(NO2)2}$ is, at least from a charge perspective, also correct. But that is the nitrite of $\ce{Q}$.