The top quark flavor changing neutral current (FCNC) processes are extremely suppressed within the Standard Model (SM) of particle physics. However, they could be enhanced in a new physics model Beyond the Standard Model (BSM). The top quark FCNC interactions would be a good test of new physics at present and future colliders. Within the framework of the BSM models, these interactions can be described by an effective Lagrangian. In this work, we study tqγ and tqZ effective FCNC interaction vertices through the process e− p→ e− W q+ X at future electron proton colliders, projected as Large Hadron electron Collider (LHeC) and Future Circular Collider-hadron electron (FCC-he). The cross sections for the signal have been calculated for different values of parameters λ q for tqγ vertices and κ q for tqZ vertices. Taking into account the relevant background we estimate the attainable range of signal parameters as a function of the integrated luminosity and present contour plots of couplings for different significance levels including detector simulation. We find the sensitivities to the branching ratios (B R (t→ q γ)= 7.5� 10− 6, B R (t→ q Z)= 3.5� 10− 5) and (B R (t→ q γ)= 8.5� 10− 7, B R (t→ q Z)= 6.0� 10− 6) for an integrated luminosity of 2 ab− 1 at LHeC and FCC-he, respectively.