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I'm writing a story in a system which has two earth like habitable planets orbiting the star in Lagrange points L4 & L5 of a gas giant similar to Jupiter / Saturn.
Is that orbit possible in the Sun like star habitable zone? Something no further than Mars assuming.
For illustration purposes: One earth like planet is in L4, the second earth like planet is in L5. The whole orbit is where Mars is now.
I'd say yes(ish), but with some caveats. It would be a very rare occurrence where everything would have to be just so. There are a number of integrations of the scenario that have been done, such as in this paper: https://onlinelibrary.wiley.com/doi/abs/10.1002/asna.200710789 or this one: https://ui.adsabs.harvard.edu/abs/2005CeMDA..92..113E/abstract
There's also the question of how these planets would form in the first place, as capture is hard to imagine taking place. Here there is a simulation of that which has been done, but it shows that the process seems to top out at planets 0.6 times Earth mass (which at least is big enough to be habitable in principle): https://www.aanda.org/articles/aa/abs/2007/07/aa6582-06/aa6582-06.html
One caveat is that the stability conditions for trojan/co-orbital bodies drastically decrease the more stuff you have in the same system, which is probably why our own system more closely follows the rule of thumb mentioned on Wikipedia, where the trojan bodies should be of asteroid proportions: https://en.wikipedia.org/wiki/Trojan_(celestial_body)#Stability
Also when I've tried putting a Jupiter sized planet with co-orbitals into an N body simulator, I found that adding in moons for the Jovian tended to disrupt either the moons or the trojans. I could only get around this by making everything smaller.
So the answer is yes but the caveats are that the co-orbitals might be limited to a bit over half the mass of Earth in terms of formation, and if there are a lot of other large bodies in the system, like other gas giants, the maximum mass for stable long term orbits goes down exponentially. You're looking at a quite empty system probably apart from the gas giant and its co-orbital planets and these will likely be smaller than Earth, under the rare scenario of them forming.
The Trojan orbits are potentially stable for a planet-sized body with low eccentricity and a sufficiently massive "Jupiter" for a period exceeding the age of the solar system. That means that if you can get your "super Trojan" into an orbit about the L4 point, then it can stay there for billions of years.
The difficulty is getting it there. The formation of the gas giant will tend to clear gas and dust from its orbit, so it is hard for a planet sized body to form at the L4 point. It is also very hard to capture a planet sized body at the L4 point. Asteroids can engage in interactions and become trapped in the 1:1 resonance orbits, but there is nothing at L4 that can pull a planet that is not in resonance into a Trojan orbit, or adjust its velocity once it is there to exactly the right velocity to be captured.
So Trojan orbits are stable, but getting there requires some handwaving.
