Is there any way to index my Github account to google search. I see that many people's account show up as soon as I type their name, but mine doesn't show up even after I type my name followed by github n search.
You can't force Google to index a page if it doesn't want to. You can suggest one to Google by using this link:
If you were an owner/administrator of the Github website you would have more options available to you by using Google Webmaster Tools
Garrulinae's answer is technically correct, but leaves out key information. Ownership verification with Google Search Console (GSC) would improve the search ranking of your Github account, repository, or website, as documented on google.com, so obviously that method is preferred. But currently it's not possible.
Many sites allow ownership verification through an arrangement or procedure worked out with Google, such as Squarespace, Medium, etc, in which you indirectly submit Google verification info for pages you control (a property, to use Google's GSC terminology). After submission these sites handle verification codes and .html files for you. Unfortunately that's not possible with LinkedIn and Github, because they are owned by Microsoft, who has no such arrangements with Google.