In Deep, deep space, there exists a star system with a single planet. This planet, named Dewel, is a large, green gas giant with a mass of about twice Jupiter's. What could this planet's atmosphere composition be to make it green?
You need methane, ammonia and atmospheric temperatures of $∼$150 K.
The color of a giant planet depends on the type of clouds dominating its upper atmosphere. These in turn depend on the temperature of the planet, as different molecules reach their gaseous phases at different temperatures. There are a few realistic potential cloud types on gas giants:
- Methane, dominant at $∼$100 K and producing a cyan-aquamarine color
- Ammonia, peaking around $∼$150 K and giving teal hues
- Water, found in atmospheres of $∼$250-300 K and producing deep blues
- Alkali metals, gaseous from $∼$500 K on up and giving the planet a reddish color
Take Uranus as an example. Though most of its atmosphere is in fact hydrogen and helium, methane clouds at the topmost layers are responsible for its green-blue appearance. If you're looking for a greenish planet, the best you're going to get is a layer of ammonia-methane clouds at the top, like Uranus and Neptune.
The teal-ish colors of Uranus and Neptune, on the other hand, are found at lower temperatures. This is because volatiles like ammonia and methane are only found farther out from a star, beyond what we call the ice line. You'll need to place this planet fairly far away from its parent star, lest these compounds be unavailable.
Chlorine won't work
Folks have mentioned chlorine; I agree with them that it won't work. Putting aside the issue of how you could get enough of it to occur naturally, chlorine's boiling point is about $∼$240 K, which is higher than the temperatures of giant planets where ammonia and methane dominate. An atmosphere with a substantial layer of chlorine-rich clouds would likely be hotter than that, in the realm where water-rich clouds could also form, making chlorine potentially not the dominant cloud constituent. Furthermore, an atmosphere rich in methane and ammonia could only exist at temperatures below chlorine's boiling point, making them incompatible.
Green atmospheres are weirdly challenging to construct. The closest in our Solar System seems to be Uranus, which is only slightly more green than Neptune's considerable blue. If you want a rich, lime or forest green, there might be problems, but perhaps taking Uranus's dose of green and pushing it as far as it can go before something breaks will be good enough?
Temperatures are somewhat relevant, since, no matter what, you're going to get majority Hydrogen / Helium atmospheres for the overwhelming majority of gas giants, and temperatures will affect what other trace elements can do. So I see four options, none of them seeming quite right:
Super Uranus, Now with More Methane
How green this would be, rather than a bluish teal, I'm not so sure.
350–900 K, with a Sulphurous Haze
Sulphur hazes tend to tinge atmospheres orange, so getting this to help with the green would be somewhat troublesome. I'd mostly want to try and mix it with the methane to see if it helps at all, but it probably won't.
Throw in a Green Gas, such as Chlorine
I have no idea how you could get enough chlorine in a gas giant atmosphere to make it look green, as chlorine is highly reactive, there's plenty of hydrogen, and if you have megatons of chlorine (all denser than the H/He atmosphere, remember), you also have the more abundant elements between helium and chlorine, which will inevitably get in the way and ruin your nice green ball of gaseous death.
1500 K:+ Make it Hot Enough for Metal Clouds
At these temperatures, sodium, silicon, and perhaps even iron compounds start forming clouds. This could give you a greenish-grey atmosphere, if everything is just right. It could also give the planet a blue halo or a comet-like tail, because at those temperatures, gases are being blasted into space at an alarming rate. I'm not sure if Mercury would have wound up like this under the "Was Mercury a gas-giant?" hypothesis, but I imagine that a gas-giant baked green would inevitably lose its atmosphere and leave a Mercury-like core behind after a few million years.
The trouble with green is that you need to both reflect green and absorb everything else, and while some substances do that at close range, scattering and black body radiation work against you in thick atmospheres. The most promising options will usually be in conflict with other necessary features of the planet. Methane is not too rare, and does greenen Uranus a little more than Neptune. Gases like chlorine are unlikely to find their way high enough to color an atmosphere, and if it does, there will be numerous more abundant elements interfering. Green particulates, or high-temperature metallic clouds can appear greenish, but those typically require the kinds of temperatures that can create metallic clouds, and that's a good way to lose an atmosphere over time.
My favored options are Super Uranus, or Metallic Clouds, as the others feel far more handwavy. And greenish-grey with a stormy aura/tail could be kinda neat, setting permitting. But I'm not sure that either is quite so green as you're looking for.
While CAE Jones already got the inorganic chemistry settled, I'd like to propose an even more unlikely biological explanation. Let's use Sudarsky's gas giant classification and take a class two gas giant with water clouds.
Now let's assume that life found it's way to this gas giant via panspermia and got stuck there. While cloud alge don't exist on Earth, it doesn't mean that they can't exist elsewhere. The fast winds which travel around the globe as uninterrupted bands and the possibly higher air pressure at the altitudes, where the clouds exist, create better connections for sky alge.
The planet probably wouldn't be entirely green. Some latitudes would offer better conditions for the sky alge than others. Expect a green equatorial region and white poles or vice versa.
As a side note, class five is a "metallic cloud" gas giant like CAE Jones suggested. Just in case your are interested in an illustration.
If you can't change the color of the room, change the color of the bulb.
This may be a bit simplistic for what you are looking for, but if green clouds are impossible under the lighting conditions we are used to (good ol Sol), change the color of the star. That way you can still use plausible gaseous compositions, but they will be an entirely different color because of the light that is available to reflect. For example, Mars would be almost black if our star were green. So if the clouds in your planet were white, and the star were green...