Both readings are valid for the kanji 紅葉
.
According to this page, the usage is as follows:
こうよう is mostly used to refer to the red/yellow autumn leaves before they fall, as well as to the scenery/images [involving them].
もみじ is mostly used to refer to the tree itself, or especially to the tree(s) that show the autumn colors
The first word can be used as a verb, the second cannot, i.e.:
〇 紅葉{こうよう}している楓{かえで} Autumn-colored maple tree(s)
× 紅葉{もみじ}している [wrong usage]
However the other way around is possible, i.e. you can call the trees kouyou when they show autumn colors.
The page then goes on to cover more related words, e.g. difference between かえで and もみじ trees and leaves.
P.S. To describe the activity of maple-tree viewing/hunting, say 紅葉{もみじ}狩{が}り and not 紅葉{こうよう}狩{が}り