The expression "a little green around the gills" refers to someone looking ill or nauseous. Etymologically, "gills" in this context is a metaphorical reference to the area of the face around the cheeks and jawline. The term "gills" originally comes from Old Norse gil, meaning a narrow valley or ravine, but in this idiom, it draws on the anatomical feature found in fish.
Fish have gills located on either side of their heads, which are essential for breathing underwater. When humans are described as being "green around the gills," it evokes an image of pallor or a sickly greenish hue that can appear on someone's face when they are feeling unwell, particularly around the lower part of their face where fish have their gills.
Thus, while humans don't literally have gills like fish do, the expression uses this imagery to describe a similar area on our faces and conveys a sense of queasiness or illness.