I am trying to search for the first person who introduced group theory into quantum field theory.
That would be my guess too.
In a classic paper published in 1939.
I would say, Wigner was mainly responsible for all applications of group theory in QM (and then QFT), whereas Weyl (from von Neumann's Hilbert spaces) introduced group methods to study the fundations of quantum mechanics.
Indeed --- any Weyl also introduced much of the modern differential geometry and group technology to the study of classical mechanics. The term "symplectic" was in fact coined by him </trivia>
Separate names with a comma.