If you are seeing distinct colors under UV light, you are not seeing UV light but fluorescence in the visible spectrum. Fluorescence of minerals will typically be monochromatic, with an intensity focused on a single wavelength. The colors in a rainbow are of that kind, and rainbows often have some appearance of unreal iridiscence in bands that just doesn't make it onto pictures of rainbows on paper or screen. It's outside of the representable color space of those media.

A camera's color vision is different from a human's one: that's why we need white balance. The camera will try to reconstruct something into sRGB color space that is supposed to look similar to the human eye as what it has been looking at. That works better for some things than others.

And monochromatic ones are the worst. While a camera does have UV filters, they usually are less discriminative than the human eye so the near-visible color band has a larger impact on the sensor than on human vision. That means that the pure monochromatic color you want to be seeing is considerably watered down by what the camera still sees as something purplish while it doesn't faze your eyes. The camera sees a lot more reflection of the UV light than it sees the fluorescence caused by it. Your eyes adapt to the dark, your cameras sight doesn't.

You could probably improve your chances by using a photographic filter that cuts off inside the visible wavelengths, like below 450nm (the usual cutoff for UV filters is at 390nm). That will somewhat impact color rendition but it will move the UV light into the dark for the camera and make it more likely that you can get a good look at greenish fluorescence without getting fazed by the UV light itself.