Asymmetric Laplace data how do we test data are asymmetric Laplace distribution? And how we convert the data that have become asymmetric Laplace distribution?
 A: You can't show the distribution from which some data are drawn is asymmetric Laplace -- failure to reject doesn't tell you the null is true. 
You can only show that (sometimes) the data are not consistent with it. What sort of alternatives are you interested in having power against? 
I note however that you chose the tag quantile-regression. If this is in the quantile regression context, the distribution of the unconditional response is not itself asymmetric Laplace (for it to be ML at least); the conditional distribution is. You can't check that assumption by looking at the data itself (the residuals may be a reasonable approximation in large samples).
My suggestion for that case would to not test at all, since that answers the wrong question. Instead I'd suggest looking at a standard Laplace QQ plot, which for large samples should show something close to a kinked line:

That would give you some sense of the extent to which a quantile regression would be close to ML, at least. However, you don't require asymmetric Laplace distribution of errors to perform quantile regression.
