Searching on a name dictionary you'll get a long long list (93) of "midori" as a girl's given name. This excludes "midori" being used as a family name or a place name.
"Midori" is not limited to the kanji for green though. It can be made up of other kanji having
名乗り (nanori - name reading) of "mi", "do", "ri", "mido", "dori" compounded to form "midori".
And yes you can use
翠 as a standalone kanji for the name Midori.
For brevity I will not list all 93:
Kana and Kanji mix:
3 Kanji compounds:
2 Kanji compounds:
There is also the possibility that a name is spelled purely in Hiragana:
In this case, writing that person's name using Kanji would be wrong.
緑野 [みどりの] (Don't worry about the meaning of "no" it's just the way the name sounds) it can be both a girl's name and a family name.
緑埜 [みどりの] is a family name instead of a girl's name.
Alternatively it could be Midori + Genitive case marker の.
みどりの本 to mean "Midori's book"
緑の本 to mean "a green (coloured) book"