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We introduce the notion of Lie invariant structure Jacobi operators for real hypersurfaces in the complex quadric 
$Q^{m}=SO_{m+2}/SO_{m}SO_{2}$. The existence of invariant structure Jacobi operators implies that the unit normal vector field 
$N$ becomes 
$\mathfrak{A}$-principal or 
$\mathfrak{A}$-isotropic. Then, according to each case, we give a complete classification of real hypersurfaces in 
$Q^{m}=SO_{m+2}/SO_{m}SO_{2}$ with Lie invariant structure Jacobi operators.
