It is very possible. Using 0/1 to indicate False/True, $x=1$ to indicate red, and $w=1$ to indicate green, the model is

$$ \operatorname{logit}(p) = \beta_0 + \beta_1 x + \beta_2 w$$

Here, we have set the reference group to be yellow. Note the following

• $ \beta_0 $ Is the log odds for the yellow category, and hence $\exp(\beta_0)$ is the odds of yellow.
• $\beta_0 + \beta_1$ is the log odds of the red group. The odds of red is then $\exp(\beta_0 + \beta_1) = \exp^{\beta_0} \cdot \exp^{\beta_1}$
• A similar argument can be made for green.

As for p-values, this depends on what the null hypothesis is. Typical logistic regression output will report p values for the Wald test that $\beta_j$ is 0. If you edit your question to be more specific about the associated null for the p value in question, I can be more precise.