To my knowledge this is not possible. In googling this real quick, everything I found also backed up that notion. Purdue OWL is one of my favorite go-to places for citation questions and they also make no mentioning of being able to do this shortcut. Everything I have read says that you need to have the quote fully cited each time.
My advice would be to write each quote out as if they are all unique sources. Then I would email your professor or talk to them next class.
If nothing else, you can always ask the professor in class next time about how you want to know if there is any shortcuts for in-text citation when dealing with a singular source for a whole paper or if you have multiple quotes from the same source that follow each other in the same paragraph.
I don't think you are entirely wrong in your notion. I remember being able to do something similar, but I can't recall if it was MLA or APA, though this may have changed since I last had to write an essay. From my research, I would go with the answer of no it is not possible.