The flame test is quite simple to do and very useful on the chemistry world, but to understand what your results mean you need to know previously a range variety of element-color relations. So, since the wavelength of the photon emitted reliess on the quantum jump done, I was wondering if, somehow, is it possible to make a fairly certain prediction of the results, based on the compound's structure. If someone knows a relation, it would be very useful.
The question is why would you want to predict that? All elements have been screened for allowed and forbidden transitions ranging over the complete visible and non-visible spectrum. You can just look it up there. There are also multiple possible transitions which lead to the often complex spectra.
And then there is the problem with the actual jump. In a simple shell-model where you only add electrons in shells this jump is just to the next level. And you can introduce Rydberg-atoms and get element specific Rydberg-constants. But I remember to have tried that for a Lithium atom and if I didn't do a mistake back then it would have required quite a high energy level to get the red emission while in reality the principal quantum number (n) doesn't even change for lithium. You can see this pretty easily if you look at the energy diagram for the famous yellow sodium emission line. It's a jump from the $3p$ to the $3s$ orbital.
But in the end, as I said, they have all been calculated and confirmed. You can just look at lists on the net where every possible transition is given with the orbitals, energy, wavelenght and something like an intensity.
And engines like WolframAlpha even use that. You can just search there for let's say sodium emission spectrum and it will give you the spectrum. And there are also websites where all spectra are given.
I'm a bit confused with your question. It sounds like you don't have such a list. I guess, as you are talking about flame colors and not spectra you are only looking for the flame color itself. This should be mentioned in most analytical chemistry books and I think there was even a Wikipedia article with pictures. But at some point this will become quite hard. Lithium vs Calcium vs Strontium is such a case where I wouldn't be too sure if I could distinguish them by their flame color without looking at spectra. In the end the observed emission will be mixture of many weak and strong emissions.