Let $X$ be a complex compact manifold. I want to understand the holomorphic tangent vector field on $X$. I know a smooth vector field can define an infinitesimal diffeomorphism of $X$ (more precisely, a one-parameter diffeomorphism group). Similarly, if we consider a holomorphic tangent vector field, will it determine to a infinitesimal holomorphic equivalence of $X$ (a one-parameter holomorphic equivalence group)?
Yes, the flow will give a 1-parameter group of biholomorphisms. See chapter 3 of S. KOBAYASHI, Transformations Groups in Differential Geometry.