The Large Hadron Electron Collider and the Future Circular Collider-hadron electron with high center-of-mass energy and luminosity allow to better understand the Standard Model and to examine new physics beyond the Standard Model in the electroweak sector. Multi-boson processes permit for a measurement of the gauge boson self-interactions of the Standard Model that can be used to determine the anomalous gauge boson couplings. For this purpose, we present a study of the process at the Large Hadron Electron Collider with center-of-mass energies of 1.30, 1.98 TeV and at the Future Circular Collider-hadron electron with center-of-mass energies of 7.07, 10 TeV to interpret the anomalous quartic gauge couplings using a model independent way in the framework of effective field theory. We obtain the sensitivity limits at 95% Confidence Level on 13 different anomalous couplings arising from dimension-8 operators. The best limit in () parameters is obtained for parameter while the best sensitivity derived on () parameters is obtained for . In addition, this study is the first report on the anomalous quartic couplings determined by effective Lagrangians at ep colliders.