In this paper, a study is carried out on the e(-) p -> e(-) gamma* p -> pW(-) gamma nu(e) production to probe quartic W+ W- gamma gamma couplings at the Large Hadron electron Collider (LHeC) with root s = 1.30, 1.98 TeV and at the Future Circular Collider-hadron electron (FCC-he) with root s = 3.46, 5.29 TeV. Production cross-sections are determined for both at leptonic and hadronic decay channels of the W-boson. With the data from future e(-) p colliders, it is possible to obtain sensitivity measures at 95% C.L. on the anomalous f(M,i)/Lambda(4) and f(T, j)/Lambda(4) couplings which are competitive with the limits obtained by the LHC, as well as with others limits reported in the literature. The production mode e(-) p -> e(-) gamma* p -> pW(-) gamma nu(e) in e(-) p collisions offers a window for study the quartic W+ W- gamma gamma electroweak bosons couplings at the LHeC and the FCC-he, which provides a much cleaner collision environment than the LHC.