Let $X\subset \mathbb P_k^n$ be a smooth quadric ($n\geq 4$). The variety of lines $F(X)$ of $X$ has dimension $2n-5$ and the incidence correspondence $$I_F=\left\{([l],[l'])\in F(X)\times F(X), l\cap l'\neq \emptyset\right\}$$ has dimension $3n-7$. There is also the incidence correspondence of $\mathbb P^n$, $$I_G= \left\{([l],[l'])\in G(2,n+1)\times G(2,n+1), l\cap l'\neq \emptyset\right\}$$ which has dimension $3n-2$. Is there a description of $I_F$ in $I_G$ as zero locus of a regular section of a vector bundle ?

$I_G$ can be represented as a degeneration locus for a morphism of vector bundles. Explicitely, if $V$ is a vector space of dimension $n+1$ and $U_1$, $U_2$ are the tautological rank 2 subbundles on the two copies of the Grassmannian, one can consider the composition $$ \phi : U_1 \boxtimes O \hookrightarrow V \otimes O \boxtimes O \twoheadrightarrow O \boxtimes (V/U_2) $$ and then $I_G = D(\phi)$. Consequently, its resolution of singularities $\tilde{I_G} \subset Fl(1,2;V) \times Gr(2,V)$ is the zero locus of a regular section.

The same approach should also work for $I_F$.