I have completed an internship under the guidance of my boss. Is it right to ask my boss to let me publish the source code in github, though the organisation I interned with will be using a modified version of it? FYI I already submitted the source code for my graduate project (It's an internship cum project) .
You must get a permission for it from your manager.
Sometimes such a request might look silly, sometimes it might make sense but chances are that your request will be rejected unless contributing to open source is in the culture of the company.
If it is something like a library for reasonable common problem, or you modified some open source project it might make sense to open source it. For example, the company might benefit from outsiders contributing to it.
But normally it has very little benefit, has some risk which makes it not worth it for the company.