Insufficient data. The number of minimally-distinguishable colors that a creature can distinguish is totally separate from the dimensionality of their color space. Humans can distinguish tens of millions of colors; birds can do the same, and if having a fourth color receptor type makes that number larger... well, that doesn't really have any practical consequence. The number is already so large that it might as well be infinite.
However, they can see completely different kinds of colors. See Ways of Coloring: Comparative Color Vision as a Case Study for Cognitive Science, Describing Non-human Vision, and What Are Tetrachromatic Colors Like?.
Exactly how the colors are organized would depend on the relative overlap in response curves between each of their four color receptor types, but they would have four physical primaries and 6 perceptual primary hues corresponding to the endpoints of 3 opponent axes output by retinal preprocessing. We can generically label those the R-G axis, the Y-B axis, and the P-Q axis.
Binary combinations of those basic hues with their non-opponents results in 12 maximally-distinct secondary colors (R+Y, R+B, R+P, R+Q, G+Y, G+B, G+P, G+Q, Y+P, Y+Q, B+P and B+Q).
So far, that is analogous to the structure of trichromatic vision, just with more boxes, but ternary combinations of non-opponent primary hues produce 8 extremal instances of an entirely new kind of hue--tertiary colors--not found in the perceptual structure of trichromatic color space (R+Y+P, R+Y+Q, R+B+P, R+B+Q, G+Y+P, G+yY+Q,G+B+P, G+B+Q). Additionally, there is not merely one non-spectral secondary color (magenta) in the fully-saturated hue space, but 3--and in general, that number will correspond to however many pairs of non-spectrally-adjacent sensor types there are (which actually works out to the sequence of triangular numbers!)
If we assume without loss of generality that R, G, B, and Q are the physiological primaries (note that the spectral locations of y and p depend on the decorrelation output for a specific set of 4 receptors with species-specific sensitivities), then the non-spectral secondaries are R+B, R+Q, and G+Q. All of the tertiary colors are non-spectral.
That covers all of the maximally-saturated hues, which exist on a color sphere rather than a color wheel. Those hues can still be modified by varying levels of saturation and luminosity, just as in trichromatic vision.
How those 26 basic hues would grouped into named categories is entirely dependent on language and culture.
Why would we worry about theoretical tetrachromatic birds, when simple Searches show there not only might be, but already are tetrachromatic people?
Tetrachromatics might be better able to distinguish combined shades or hues or what terms you choose but is that any use?
Isn't the real point how many shades or hues can be distinguished in each base colour?
Are you assuming the same graduation apply to each of four colours, as to three?
– Robbie Goodwin Feb 29 '24 at 21:49