 Solution

Yes. Let $g$ send the first flag to the second and write $d=\det g\ne 0$. In an adapted basis of the target flag, scale its first vector by $d^{-1}$ and fix the others. This defines a flag-preserving map $h$ with determinant $d^{-1}$. Then $hg$ still sends the first flag to the second and has determinant one. Thus $\mathrm{SL}(n,K)$ is transitive as well.