Folder searching already works in Alfresco. What do you expect to find? Do you even have a folder that would match that search input? Dou expect folders to turn up as live search results? I believe that part is not the intended functionality - but I believe you should see folders as the result when you hit enter and load the search result screen...
But When I enter my folder name to search text, I can not see any folder in search result list. There was only documents,people or sites. For example I have a folder called "test1". I want to search this folder, nothing comes to search result about that. (like image)
Like I said - the live search results will never show folders because they have been designed to only show documents, sites and people (that's why these sections have been named that way). In order to get the actual search result page you just have to hit "Enter" to go to the full search result page.
You are free to open an enhancement request in the Alfresco JIRA. Beware though that the Alfresco Share user interface currently is not a short-term strategic priority for Alfresco, so any enhancement requests will likely not be implemented for a while. Of course you could also implement the necessary change and provide a pull request on the GitHub repository for Share, and reference that in your enhancement request - that may speed things up though I would not guarantee it...
(Potentially, an enhancement might also require a change in the Aikau project)