To answer the question, as qualia cannot be compared from person to person, there is in principle no way to show that they differ (or are the same) objectively. A sort of relative comparison of qualia within an individual (as you propose) will also not get you there, as it may be that the individual is simply more sensitive to changes along the gradient, not that they are experiencing a new color, and that's assuming we are able to use an objective comparison scale which we aren't.
Here's another way of looking at it: psychologically, we don't perceive yellow as a mixture of red and green, the way we perceive purple as a mixture of red and blue. Yellow isn't "perceived" as a mixture of anything in our minds -- it's perceived as primary. If you say "it's kind of a reddish-green", nobody is going to think, "oh you mean yellow!" While orange is perceived as a mixture of red and yellow, for example.
Remember, I'm not talking about what's happening physically with wavelengths -- I'm talking about psychological perception of colors.
The L cone is actually most sensitive to the yellow wavelengths.
There is an African tribe that can see more greens and browns, due to their genetics and it has been proven for them.
Don't quote me on this, but I believe as a whole qualia has been proven. That is, we all do experience the world differently, but we can both point at the same thing and give it a same name. Words like 'red' are just pointers to the colors we see, and everyone sees it differently. This, ofc, doesn't guarentee we can see more or less levels of detail, just that we what we do see can be different.
You can enjoy reading: https://en.wikipedia.org/wiki/Qualia
If they did this then wouldn't they necessarily also confuse some colours that trichromats can distinguish?
That wouldn't prove, but it would strongly suggest, that they're seeing the same qualia as the rest of us in the end, just mapped differently. Particularly if we could find "symmetries" -- i.e. for each area of new distinction they see, there's a corresponding area of distinction that collapses.
Let me explain: our eyes perceive RGB, but we don't in consciousness -- the RGB is mapped somewhere, long before consciousness, further in the brain to more of a perceptual RYGB -- a two-dimensional space of warm/cold (RY/GB) against "lighter/darker" for lack of a better term (YG/RB). See [1] for more info. Psychologically yellow functions as a primary color even though we don't have a rod for it. (In reality it's much more complicated than this, since we perceive saturation, brightness, etc. but this is a valid simplification for current purposes.)
So it's entirely possible that tetrachromats map their four rods to the same RYGB box the rest of us perceive as qualia, only with a different mapping. Then, they can still distinguish between colors everyone else sees as the same (because different spectra), but they'll still perceive them as the same color qualia the rest of us perceive -- colors in the world will just be mapped slightly differently. The closest (but imperfect) analogy I can think of is wearing polarized glasses -- same color gamut, but certain things "compare" differently now.
Or, does tetrachromacy somehow also change the perceived-qualia RYGB model itself? So there's, say, a 3rd axis beyond warm/cold and "lighter/darker"? Or the span of the two axes somehow becomes wider to accomodate more information? Or something else?
Hope this is clear. Perhaps it could be tested psychologically, though -- simply by asking participants to describe the colors they see qualitatively, to see if the words they use in color comparisons (e.g. warmer/colder) are the same as those used by the rest of us... or if there's any new "feeling" component involved.
[1] https://en.wikipedia.org/wiki/Opponent_process - "...the cells were widely called opponent colour cells, Red-Green and Yellow-Blue. Over the next three decades, spectrally opposed cells continued to be reported in primate retina and LGN."