The event-triggered scheduling of network transmissions has found many applications in engineering tasks operated in cyber-physical systems for its competitive advantage of system resource exploitation. This paper investigates the distributed gradient method for large-scale convex constrained problems with event-triggered consensus protocols. We show that the convergence can be ensured provided that the event-triggering threshold bound is square summable, and the stepsize satisfies specific conditions that are characterized by the Lipschitz constant of the gradient and the spectrum of the mixing matrix associated with the network topology. Stronger convergence results are derived for the strongly convex case, i.e., the local estimate of the minimizer linearly converges to the minimizer until reaching an error floor whose magnitude is shown to be proportional to the stepsize if the triggering threshold bound linearly converges. Comprehensive numerical experiments are conducted to verify the correctness of the theoretical results and advantages of the proposed algorithm over existing ones.