The commentaries on the Rambam discuss it, with regards to the Me'il (which appears to have had four corners). If there were fringes attached, it's missing from any documentation we have about them!
Minchas Mordechai al haTorah discusses this question and several proposed answers, but the strongest one appears to be that when the Torah says "put fringes on the your four-cornered garments", that applies only to personally-owned garments. The Kohen Gadol's garments were considered Temple property and not personally-owned.
At any point the Kohen was wearing non-personally-owned garments, he wasn't expected to have fringes on them. When his shift was over and changed out of them into his personal clothes (which in ancient times, usually had four corners), those would have had tzitzis.
We don't want to be people who "avoid" a mitzva, and thus today we wear four-cornered garments so we can put fringes on them. However, if someone found themselves in a situation where for legitimate reasons they couldn't wear a personally-owned four-cornered garment today, it's not that they failed their obligation "though shalt wear a fringed four-cornered garment"; it's that they didn't get a chance to connect to G-d via "if you wear a four-cornered garment, put fringes on it." So if the kohen gadol had a particularly busy day in the Temple and couldn't wear personally-owned clothes at any time from dawn till dusk, I doubt he'd be taken to task for it.
(Nor could a kohen on-duty wear a personally-owned tzitzis garment in addition to his standard uniform. While serving in the Temple, they have to wear the listed clothes, no more no less.)
A similar question comes up with regards to Tefilin. Rambam explains that there was room below the turban to wear Tefilin shel rosh; however there was no way to wear the shal yad -- both the tefilin and the ketonet's sleeve must be worn directly on the skin of the arm -- they're not mutually compatible! When a kohen was off his shift he could wear the shel yad.