Your question seems to be equivalent to fixing $V$ and an $(n-r)$-dimensional subspace $U$ and then asking how many $r$-dimensional subspaces $W$ have $\{0\}$ intersection with that $U$. The number is (unless I've made a silly mistake) $$ \frac {(q^n-q^{n-r})(q^n-q^{n-r+1})\cdots(q^n-q^{n-1})} {(q^r-1)(q^r-q)\cdots(q^r-q^{r-1})}. $$ Here the numerator counts the number of bases $\{b_1,\dots,b_r\}$for such subspaces $W$. The first vector $b_1$ can be any vector $\notin U$; then $b_2$ can be any vector not in the space spanned by $U$ and $b_1$; etc. Similarly, the denominator counts the number of bases for any single $W$, so the quotient counts the $W$'s.

EDIT: This answers an earlier version of the question:

Your question seems to be equivalent to fixing $V$ and an $(n-r)$-dimensional subspace $U$ and then asking how many $r$-dimensional subspaces $W$ have $\{0\}$ intersection with that $U$. The number is (unless I've made a silly mistake) $$ \frac {(q^n-q^{n-r})(q^n-q^{n-r+1})\cdots(q^n-q^{n-1})} {(q^r-1)(q^r-q)\cdots(q^r-q^{r-1})}. $$ Here the numerator counts the number of bases $\{b_1,\dots,b_r\}$for such subspaces $W$. The first vector $b_1$ can be any vector $\notin U$; then $b_2$ can be any vector not in the space spanned by $U$ and $b_1$; etc. Similarly, the denominator counts the number of bases for any single $W$, so the quotient counts the $W$'s.