No. There is a difference between a page break and a paragraph break. Write your story using standard paragraph breaks. Indent them as necessary. What some newbies do is to break up the actual manuscript into pages and pages. (. If they think their book is 32 pages, they turn in 32 sheets of paper with their story on it to the publisher.) Typical manuscripts don’t do this. The story is written according to standard paragraph forms (unless its poetry or something, in which case poetry formats like stanzas are used) and the page breaks (when published) are not indicated.
If you have never taken a psychology or social science class before, then you are probably accustomed to using a different style guide such as MLA or Chicago style. New college students are often surprised to find that after spending years having another formatting style drilled into their heads, many university-level classes instead require APA style. It can be a difficult transition, especially if you have to bounce back and forth between different styles for different classes. Getting a solid grasp of the basics and bookmarking a few key resources can make learning this new format a bit easier.