I am afraid that it is currently not supported to retrieve user's time zone through api . There is no out-of-the-box api to get user’s time zone. You could add your request for this feature on our UserVoice site , which is our main forum for product suggestions.
Apart from the negative answer, I think what you want is a good idea! So I post a feature request here for you. You can vote and add your comments for this feedback. The product team would provide the updates if they view it. Thank you for helping us build a better Azure DevOps.