Let $G_2$ be one of the exceptional simple Lie groups and $\mathfrak{g}_2$ it's Lie algebra. On pges 355/356 in Fulton & Harris' Representation Theory is shown that the stadard representation of Lie algebra $\mathfrak{g}_2$ is seven-dimensional irrep $V = \mathbb{C}^7$.
In the Chapter 23.3 on homogeneous spaces, page 391, is discussed how irreps of $\mathfrak{g}_2$ give rise for parabolic subgroups $P \subset G_2$.
The action of the Lie algebra on the standard rep $V$ induces an action on projectivized space $\mathbb{P}V \cong \mathbb{P}^6 $. Via exponentiation this induces an action of $G_2$ on $\mathbb{P}V$. Let $p \in \mathbb{P}V$ be the line corresponding to the eigenspace of the highest weight $\lambda_V$ of $V$, because $V$ is irrep.
The correspondence tells us that the parabolic group $P_{\lambda} \subset G_2$ corresponds to the stabilizer of $p$ and thus the orbit $G \cdot p$ equals $G_2/P_{\lambda}$.
The part I not understand: the book claims then that the orbit $G \cdot p$ can only be a quadric hypersurface, since it is homogeneous. Why that's the case?
The way the argument is phrased suggests that since the orbit $G \cdot p$ is a hyperplane of $\mathbb{P}^6 $ which is by construction a homogeneous space, one should conclude that it must be a quadric surface. This conclusion is not clear me.
I'm not sure if it helps but on pages 355/356 one obtained as result of the careful analysis of properties of $V$, that the action of $\mathfrak{g}_2$ on $V$ preserves a quadratic form. Could one maybe use somehow this insight to conclude that $G \cdot p$ is given as quadric hypersurface of $ \mathbb{P}V$?
A remark: as @hm2020 noticed one should expect that it's possible to use immediately the classification result for homogeneous spaces in terms of their line bundles. But I'm not sure that the authors had this argument in mind, since the book is relatively self contained and this proposed approach is not discussed there, that's why I doubt that the autor's would conclude it from such a deepcresult without loosing any couple of words about it. Moreover the book's style is characterized by attepts to unravel the geometric picture. Therefore I hope that the proposed claim that the orbit of $p$ is a quadric hypersurface can be somehow directly concluded using only the methods the book provide.