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 than the one you suggested. For details see formula (2.8) in this paper.