The Hodge-DeRham operator whose index  on closed manifolds is the Euler characteristics admits *local* boundary conditions  on manifolds with boundary, Dirichlet or Neumann type. That is not the case  with the Dolbeault operator that does not admit  *local* boundary conditions. (This is a rather nontrivial fact observed  50 years ago by Atiyah and Bott  and involves some $K$-theory.)  

In the Dolbeault case  one has to use the Atiyah-Patodi-Singer boundary condition. This is a *non-local* condition. The resulting formula  for the index of this boundary value problem is a bit more complicated that the one  you suggested. For details see formula (2.8) [in this paper.][1]


  [1]: http://www3.nd.edu/~lnicolae/cobord-degen.pdf