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We study the classifying problem of immersed submanifolds in Hermitian symmetric spaces. Typically in this paper, we deal with real hypersurfaces in a complex two-plane Grassmannian $G_2\left({ℂ}^{m+2}\right)$ which has a remarkable geometric structure as a Hermitian symmetric space of rank 2. In relation to the generalized Tanaka-Webster connection, we consider a new concept of the parallel normal Jacobi operator for real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ and prove non-existence of real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ with generalized Tanaka-Webster...