Are these moments estimators asymptotic unbiased? In this paper, authors consider method of moments of fitting Gumbel distribution:
We know that maximal likelihood estimators are asymptotic unbiased. But are these moments estimators asymptotic unbiased? Or consistent estimators?
Can we find the reference to say these two estimators are asymptotic normal?

Yes, they are.
Equation 4 in the paper says
$$E[X]=\mu+\gamma\sigma$$
and
$$\mathrm{var}[X]= \pi^2\sigma^2/6$$
and rearranging these gives the formulas for $\mu$ and $\sigma$ as functions of the true moments
The sample moments are consistent for the true moments, so these smooth functions of the sample moments are consistent for the same functions of the true moments
