The total numbers of mutations on different branches of gene tree are independent or not?
Here is a simple answer to my question. The reason two numbers of total mutations accumulated in two divergent branches are not independent of each other is because they experience the same amount of coalescent time.

While the stationary Poisson mutation processes are independent from each other as long as they happen in different branches of a genealogy, they are likely to produce a similar number of mutations if two such processes happen together for a same amount of time. Thus, the non-zero part of the covariance between $k_{BC}$ and $k_{BD}$ is not from mutation processes themselves, but from the shared coalescent time $T$.