Just a question; Is it proper, when quoting, to quote a cartoon character??

Are you planning on quoting a cartoon character in your PhD thesis or somewhere else?

No. Forget constructing a context and answer the question generally.

It depends on what it is that you are writing about. Say , for example, you are just telling a story about something and you used "what's up doc" in the story line about two doctors at work. Then I would say you don't have to. If you were talking about the cartoon characters themselves then you should. Again if you were writing about Bugs Bunny or what Bugs said, then you should use that in quotation marks.

Thank-you cosmic...

Yes, quoting a cartoon character is valid.
Just give the title of the cartoon.