The Method of Moments' estimators for a beta distribution $B(a,b)$ are $$\hat{a}=m\frac{m(1-m)-v}{v},$$ and $$\hat{b}=\hat{a}\frac{1-m}{m},$$ where $m=\bar{X}$, and $v=\frac{n-1}{n}S_n^2$.
Now, for checking whether estimator $\hat{b}$ is consistent, we have to prove that it converges in probability to the true parameter $b$. That is, $$\lim_{n\rightarrow \infty}\mathrm{P}\left( |\hat{b}-b| > \epsilon \right)=0.$$
Using Chebyshev inequality, we get $$\mathrm{P}\left( |\hat{b}-b| > \epsilon \right)<\frac{\mathbb{E}\left[ (\hat{b}-b)^2 \right]}{\epsilon}.$$
Therefore, we only need to prove that $$\mathbb{E}\left[ (\hat{b}-b)^2 \right]\longrightarrow 0$$ for $n\rightarrow 0$.
Here, I am stuck as I do not know how to compute the expectation.