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I can't recall the name of a novel where a young human woman takes care of a group of children who may have psychic powers but the government does not know what their abilities are or when they will manifest. When the woman identifies that a child has an ability she notifies the authorities who take the child away, most like to be dissected for experimentation.

After the first child with an ability is taken away the young woman decides to protect the rest of the children. Most of the children display abilities. The book may be set on a starship due to some of the references.

When the young women goes out to socialize she wears a dress of twigs and leaves as she does not have a full body tattoo which is the current vogue.

There was an alien, a green being that may have been like a pet. The book itself had a green cover, and may have the word "green" in the title.

Perhaps Green Is for Galanx, a 1980 novel by Josephine Rector Stone. The cover has quite a bit of green in it.

Excerpt from a review by Baird Searles in Isaac Asimov's Science Fiction Magazine, October 1980, available at the Internet Archive:

Willy's World is a self-contained, artificial satellite that has been
wandering around the universe for several generations (shades of
Space: 1999 and Lost in Space!). It is a despotism that's not all that
benevolent, and a group of mutant children with psi powers and
their guardian, Ilona, fear that the children will be killed in order
to analyze their talents for adaptation to android mentalities. When
the chance comes, the group, accompanied by the shape-changing
Galanx which they've more or less adopted as a pet, escapes to a
new planet. There they must cope with the strangeness of a natural
world, and escape the inevitable pursuit.