In APA, do we italicize the names of wars and treaties?
The APA guidance doesn't say exactly.  Gives a list of what to italicize and examples of what not to.  I would definitely opt for not doing it though.  The implication of the directions are much more to only italicize what you need to, not italicize everything not forbidden.

https://academicguides.waldenu.edu/writingcenter/apa/other/italics

Note also, that I'm really unaware of ANY style (just checked Harcourt Brace and Merriam Webster, as well as personal experience) that recommends italics for wars or treaties (or legislation).  I have only seen treaties in italics when the entire treaty (not just a word within it) is cited in a foreign language (often French, Latin or German), but then you are treating it as foreign text, not because it's a treaty.  Oh...and don't italicize a single foreign word within a war or treaty.  Keep entire thing normal text (but capitalized first letters, like a book).  I have never seen wars italicized--perhaps you are wondering because ships are?

Note though that court cases are often italicized.