No, it is not correct to say that the electron field and the gluon field interact with each other via the electromagnetic field. The electromagnetic field is associated with the interaction of electrically charged particles, such as electrons, through the exchange of photons. However, gluons are the mediators of the strong nuclear force, which binds quarks together inside protons, neutrons, and other hadrons. The gluon field is responsible for the interactions between quarks and the strong force, not the electromagnetic force.
In quantum field theory, which describes the fundamental forces and particles, each type of particle is associated with a corresponding field. The electron field is associated with the electron, and the gluon field is associated with gluons. The electromagnetic field is associated with photons, which are the particles that mediate the electromagnetic force between charged particles. The electron field interacts with the electromagnetic field through the exchange of photons, while the gluon field interacts with the strong force through the exchange of gluons.
So, the correct statement would be that the electron field interacts with the electromagnetic field via the exchange of photons, and the gluon field interacts with the strong force via the exchange of gluons.