My answer starts off just like Emerton's answer above; you want the $G$-orbits on $G/B\times G/B$. But now, I diverge from Emerton to say that $G/B$ is the space of full flags $F_0\subset F_1\subset \dotsb F_n$, where $F_i$ is of dimension $i$. Our problem is to determine the number of orbits of pairs of flags under simultaneous translation in $G$.

Suppose $F_\bullet$ and $F'_\bullet$ are two such flags, then one can first show that the dimensions $\dim(F_i\cap F'_j)$ completely determines the orbit of the pair; for if $E_\bullet$ and $E'_\bullet$ are another pair of flags such that $E_i\cap E'_j$ has the same dimension as $F_i\cap F'_j$ for all $i$ and $j$, then it is possible to construct an element $g\in GL_n$ which takes $F$ to $E$ and $F'$ to $E'$ (for example, by choosing suitable bases).

Write $d_{ij}$ for $\dim E_i\cap E'_j$. Let $w_{ij}=d_{ij}-d_{i-1,j}-d_{i,j-1}+d_{i-1,j-1}$. One may show that $w_{ij}$ is a permutation matrix and that the $d_{ij}$'s can be recovered from $w_{ij}$'s (actually, $w_{ij}$ keeps track of when the jump from $0$ to $1$ happens in the filtration of $E_i/E_{i-1}$ induced by $E'_\bullet$). Also, every permutation matrix arises in this way, for example, from the pair $E_\bullet, w\cdot E_\bullet$, where we now think of the permutation matrix as an element of $GL_n$.