Two is a hard cap on number of conflict actions. You could justify talents for lower difficulties to actions or lower cost for counterstrike or penalties to enemies’ attacks (your defense with your extra weapon) or extra minion slaying talents though.
Well, we know from A Princess of Mars that they can dual strike with at least two, as they do so in text.
We also know from same source that Tharks are all ambidextrous.
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Hullo, Finarvyn,
While what you say is true about players and stuff, the fact of the matter is I have a power gamer who’s been doing this for decades. It doesn’t bother me for the most part, other than the fact that she asked the thread’s title question, and I came here to see what folks had to say. 
As for the Tharks, A Princess of Mars makes it very clear that they are capable of wielding two weapons at once and that they are ambidextrous.
JohnK
Hullo, Werlynn,
Yep, technically you’ve got the two Conflict actions and the counterstrike. But that’s not the same thing as using all four arms to attack or defend or whatever, which is what the player in my group was getting at to begin with.
That said, to do what she wants is a Grade 8 Talent. (I worked out a rough write-up for it.) 
JohnK
Hullo, Aramis,
Awww, nuts… you beat me to it!
JohnK
From the game perspective that just means they’re more likely to have talents related to their 4 arms the same way humans are more likely to have talents related to their might.
That’s the primary peril for power gamers in a squishy rules light system—they have less to grab on to to fiddle with.
Hullo, Werlynn,
Yes, that’s the neat thing about the John Carter of Mars system and the Talents, that one can model the green Martian four-handedness with Talents easily enough - though costly, depending on what the player will want.
And yes, the power gamers do have less to get their hands on and fiddle with, but that’s another strength to me of the game system. The player, as far as I’m concerned, will just have to learn to deal with it. Whether she takes a green Martian character or not.
JohnK
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MY wife Rach came up with Four Armed Deadly Pistoleer for her Thark Adventuress. It cost her 3 points.
I’m new to the game and the forum, so please forgive the archaeology!
Based on personal headcanon from re-reading the books, the upper pair of arms are the equivalent of our dominant hand, while the Green Martian’s lower arms are their non-dominant ones. So yes, they CAN fight with all four arms, but they’re not as strong or accurate with their non-dominants. In game terms, creating a skill would be the way to get around that.