If our eyes were just a little different, we might see green when looking up at our sun. Because they aren't, we'll never see a green star no matter where we look. However, alien eyes, when they come to our solar system — or if they're already here! — might be able to see green twinkling, and here's why.
Ever heard of a blackbody? It's a body that is able to take in all kinds of radiated energy, and then spit it all back out. It radiates energy at all frequencies, but it radiates those frequencies in a very specific way, depending on its temperature. And extremely hot blackbody would radiate most of its energy in the ultraviolet spectrum, less of it in the visible light spectrum, and even less in the infrared spectrum. A coolish blackbody would radiate most of its energy in the infrared spectrum, with slightly less radiating in the visible light spectrum and even less in the ultraviolet. One between the two would radiate most of its energy in the visible light section of the spectrum, while radiating less at either end.
All blackbodies spread their energy radiation along a certain curve. As you can see this peaks at a specific temperature, drops off sharply at the temperatures above its peak, and tapers off more gently over the temperatures below its peak. Although there are only a few lines on the graph, the blackbody can emit at any temperature. Each of those temperatures would emit frequencies distributed along those curves, with a relatively gently-sloped bell at the peak.
That shouldn't be a problem for stargazers. There should be plenty of stars that peak right in the green area of the visible spectrum. And there are. The problem is, our eyes can't see them. Our eyes have cones that can see three different colors — blue, red, and green. Mixing those three basic colors together in different proportions gives us the many different colors we see. A combination of light from the three colors together lets us see white. Taking out most of the blue frequencies of light will let us see the orange, yellow, and red light. Taking out most of the red will let us see bluish shades of light. And taking out the green will let us see purple.
If you follow the blackbody emission curve for any of these, you can see that shifting it up so that it peaks at blue will let the curve taper off a great deal before it gets into the red spectrum, and so the blackbody will not emit much red light. As a result, we'll see bluish stars. Shifting the curve down so that it peaks at the red end of the spectrum will let the curve drop off steeply above it, so that the star will not emit a lot of blue light. The star looks orange or red. But the green peak is right in the middle of the spectrum. If you shift the curve so that it peaks right in the green section, you'll still get a lot of red light and a lot of blue light being emitted. Those lights combine with each other and we see white light, and not green.
Now, it's possible that we'll see a star through a gas cloud that filters out red and blue light and lets us see it as green, but that will be much the same as seeing it through green lenses. The light will be, from our perspective, dyed green. But aliens might not be as limited, when it comes to light, as we are. If their eyes aren't as sensitive to red and blue as they are to green, or if they'll be able to shut off certain "cones" in their eyes at will, they might be able to see the greenness in stars.