As flag variety or a homogeneous variety is a quotient $\Sigma=G/P$ of a reductive Lie group $G$ by one of its parabolic subgroups $P$. The subgroup $P$ fixes a flag of subspaces of standard representation $V$ of $G$. There is an embedding of projective varieties $\Sigma\subset \mathbb P V_{\lambda}$, where $V_{\lambda}$ is some highest weight representation of $G$.

For the exceptional Lie group of type $G_2$, if we consider its highest weight representation for highest weight $\omega_2$ then we have an embedding of a homogeneous variety $\Sigma\subset \mathbb P V_{\omega_2}$. Since its a subvariety of Gr(2,7), which can easily be seen to be a "flag variety", so we can some how realise this $G_2$ variety as a flag variety.

If we consider an exceptional Lie group $F_4$ and take its highest weight representation with highest weight $\omega_1$ then $V_{\omega_1}$ is 26 dimensional and we have an embedding of $F_4$ homogeneous variety $\Sigma ^{15}\subset \mathbb P V_{\omega_1}$, which is a codimension 10 embedding. My question is that how can we realise this variety as a "flag variety" or is it also a subvariety of some other standard flag variety?

isa flag variety of $F_4$; it's a quotient of $F_4$ by one of its maximal parabolic subgroups. Are you asking whether there is some classical Lie group $G$ that contains $F_4$ and that acts on $\Sigma$ in such a way that $\Sigma = G/P$? (I think this question was essentially asked before on MO, but I can't find it now.) In that case, the answer is 'no'. Also, is your $F_4$ complex or are you dealing with one of the real forms? – Robert Bryant Oct 27 '11 at 22:41