Hey Mona, how are you?
In the sentence you provided (''She's got a few wrinkles and she's going gray''), the expression ''going gray'' could be replaced with ''becoming/getting old''. The expression ''go gray'' is usually used to informally say that a person is aging or is far along in years.
The usage of the word ''gray'' in this case alludes to the state of a person when they are lacking in cheer and are becoming gloomy and old with the passage of time - as if the years had leeched (away) at their brightful/colourful hair (it is now becoming grey), and clear and soft skin.
Hope that helps! :)