If one is typing the name of the chapter inside a story, do they put quotes around it? If not, how is one indicated?
This depends largely on your purpose for writing. For example, if you are writing a novel, fictional work, or other non-peer reviewed content, it generally becomes in issue of personal preference.
If you are writing an extended article, textbook, research paper, or something else of an academic nature, your best option would be to consult with the style guide for the specific discipline of your content.
Some common ones for reference: APA Turubian (Chicago) MLA AMA CSA ACS