  We have studied the induced one-loop energy-momentum tensor of a massive
complex scalar field within the framework of nonperturbative quantum
electrodynamics (QED) with a uniform electric field background on the
Poincar\'e patch of the two-dimensional de Sitter spacetime
($\mathrm{dS_{2}}$). We also consider a direct coupling the scalar field to the
Ricci scalar curvature which is parameterized by an arbitrary dimensionless
nonminimal coupling constant. We evaluate the trace anomaly of the induced
energy-momentum tensor. We show that our results for the induced
energy-momentum tensor in the zero electric field case, and the trace anomaly
are in agreement with the existing literature. Furthermore, we construct the
one-loop effective Lagrangian from the induced energy-momentum tensor.
