When reporting the results from a Fisher's exact test, do I just report it like (fishers exact, p = .02) or is there a test statistic for Fisher's exact test?
What exactly you will report in writing up your results will depend on your field and the journal you are submitting to. We cannot exactly answer that. However, your suggestion seems reasonable. Note that Fisher's exact test does not have a test statistic (like $t$ in a $t$-test): What is the test statistic in Fisher's exact test?.