Have you seen the user-written command ivqte - Instrumental-variable quantile treatment effects? There is an accompanying paper that has the link below.

http://www.stata-journal.com/article.html?article=st0203

Many thanks to Ariel and Umar. I do notice ivqte command in STATA and also the article given by Ariel. However, it seems that the ivqte command cannot use for the case when endogenous variable is not a binary variable. 

Umar's advice might be a good idea. Can you be helpful to show some article which use the procedure to do IV-QR? Thanks again.

Try the "cqiv" command at: https://ideas.repec.org/c/boc/bocode/s457478.html

There is an option for NOT using "censored" observations.

Many  Thanks, prof. Huang. Your answer helps me out.

I guess that Umar's suggestion may produce not consistent estimates and not reliable confidence interval . The same issue was already discussed in a very long thread on statlist long time ago. You may want to take a look: https://www.stata.com/statalist/archive/2008-05/msg00897.html

I agree with Laura. OLS at the first stage + QR at the second is not reliable in CIs but above all it yields inconsistent estimates. You could try the Chernozhukov-Hansen method for IVQR. CQIV.

Unfortunately the IVQTE by Froelich and Melly does not seem to work on all databases. Furthermore if I remember well, it only works on binary endogenous treatment variables.

can any one help me how to handle the problem of endogeneity in Bayesian Quantile regression?

please study different papers on quantile regression

Can any one suggest how to test the validity of the instruments in quantile regression such as the Sargan statistic in instrumental variable regression.

Do a falsification test

Different papers cannot show the command for instrumental variables using quantile regression

Ho-Chuan Huang how to run spatial quantile regression?