You'll need to look up the hunting seasons on the regulations pages (those 2 big pdfs, on the page Chumley linked to, one has elk, the other has deer). Contrary to what Chumley said, you're pretty much looking for deer/elk seasons on the Highline. There are hunters out for other game, but numbers are way lower and they're spread over longer seasons.
To find the game units, you used to be able to check this map: https://hikearizona.com/map.php?QX=4230
but it looks like the recent map updates made it harder to see which unit is which. Double-click in the unit, and it will show the unit number to the left. Anyone know if there is a way to get those labelled automatically? When you double-click, it automatically zooms in, which makes that an unwieldy way to see the units.