That quantum analogue is known as the(*) Ward-Takahashi identity.
An extension of this theorem for holonomic non-conservative systems states that this theorem holds if there exist solutions to the Killing equations. In addition to that extension from Djukic and Vujanovic, another version of this statement relates off-shell correlation functions in momentum space of an external photon and several electrons to a series of smaller amplitudes that omit the influence of the photon. That quantum analogue is known as the(*) Ward-Takahashi identity. A derivation of this theorem yields a continuity equation for a namesake current whose "charge" is a quantity such as energy or angular momentum. For 10 points, identify this theorem that states that a physical system with a continuous symmetry will have a corresponding conserved quantity.
An extension of this theorem for holonomic non-conservative systems states that this theorem holds if there exist solutions to the Killing equations.
Noether's First Theorem