If the only time we see the use of the extra arms as legs is as newly hatched babies, it doesn’t seem they gain any great advantage from using the "intermediary limbs" as a second pair of legs. Why do they never take advantage of this as adults? Maybe they derive no game-mechanical benefits for doing so. Maybe they lose some flexibility in growth to adulthood and can’t effectively use their middle limbs as a second set of legs. Maybe it just takes too long to switch modes as joints twist and get stiff from one configuration. Regardless, it doesn’t seem four legged movement is a hallmark of the race. Maybe some unusual Green Men can still switch between orientations, taking "Semi-Quadruped" as a feat, but if that is its only appearance in the source material, it hardly seems necessary as a racial trait.
When John Carter sees the newly hatched Green Martians he thinks they have six legs, suggesting the babies are crawling on all sixes. He later learns they have "two legs and two arms, with an intermediary pair of limbs which could be used at will either as arms or legs". While it's possible to interpret this as saying
only infants have these "intermediary limbs", a couple of passages later in the book indicate adult Green Martians have them as well.
Firstly, later on in
A Princess of Mars, John Carter has an encounter with White Apes, who are described as follows:
The creatures were about ten or fifteen feet tall, standing erect, and had, like the green Martians, an intermediary set of arms or legs, midway between their upper and lower limbs.
If only infant Green Martians could walk on all fours, wouldn't the above passage use "like the baby green Martians" or something similar? It'd be like describing an animal as "crawling on four limbs, like a human" because human babies move on all fours.
The clincher is this description of Green Martians in A Fighting Man of Mars:
The terrible green warriors of Barsoom are the hereditary enemies of all the other races of this martial planet. They are of heroic size and in addition to being equipped with two legs and two arms apiece, they have an intermediary pair of limbs, which may be used at will either as arms or legs.
In conclusion, Green Martians move on all six of their limbs when born, then learn to stand on two or four limbs as they mature, much like a human baby crawls before it walks.
However, the question asked was can Green Martians run on all fours? As I said in my previous post, I don't remember them ever being described doing so. A search of the books only found a reference to White Apes running on all fours but I couldn't find and mention of Green Martians running on a particular number of legs.
In
The Gods of Mars, the second book in the series. John Carter and Tars Tarkas are fighting a bunch of Plant Men. Hundreds more Plant Men appear in the distance, together with:
some strange new monsters which ran with great swiftness, now erect and now upon all fours.
These creatures turn out to be White Apes.
Now, at last, I saw the nature of the other monsters who had come with the plant men in response to the weird calling of the man upon the cliff's face. They were that most dreaded of Martian creatures—great white apes of Barsoom.
In combination, that's a clear statement that White Apes can run "upon all fours". The
The Gods of Mars describes the White Apes' limb arrangement as follows:
They stand fifteen feet in height and walk erect upon their hind feet. Like the green Martians, they have an intermediary set of arms midway between their upper and lower limbs.
The wording is very similar to the description of White Ape limbs in
A Princess of Mars, except it describes the intermediary limbs as "arms" instead of "arms or legs". However, since
The Gods of Mars has already described White Apes running on all fours it seems pretty obvious the "intermediary arms" can also serve as legs.
Since Green Martians have the same limb arrangement as White Apes and can also use their "intermediary limbs" as legs, it seems plausible they might be able to run on all fours too.