Let $p$ and $q$ be positive integers such that $\frac{2011}{2010} = \frac{p+1}{p} \cdot \frac{q+1}{q}$.
Then $2011pq = 2010(pq + p + q + 1)$ i.e. $pq = 2010(p + q + 1)$.

From the last equation we find
$$p = \frac{2010(q+1)}{q-2010} = \frac{2010(q-2010)+2010 \cdot 2011}{q-2010} = 2010 + \frac{2010 \cdot 2011}{q-2010}.$$
Since $p$ and $q$ are positive integers, it follows that $q - 2010$ is a positive divisor of $2010 \cdot 2011$. Each divisor of $2010 \cdot 2011$ corresponds to exactly one pair $(p, q)$. Since $2010 \cdot 2011 = 2 \cdot 3 \cdot 5 \cdot 67 \cdot 2011$, the number of its divisors is $2^5 = 32$. Finally, since the pairs $(p, q)$ and $(q, p)$ determine the same representation, the number of required representations is 16.