The refer program does not support APA-style referencing out of the box. However by modifying macros like .][ and strings like [T, [A and [O, one can change the behaviour of refer.
Hans Bezemer wrote the apa.tmac macro set to adapt refer. From his own words, this set is far from perfect but it works for simple references (2022-10/msg00012).
Using apa.tmac involves having installed awk and specifying if an entry is a book or an article with the '%t' field marker.
After getting the macro set from the mailing list (ApaGroff.tar):