In this paper, we study the numerical solution of the time-fractional telegraph equation on the unbounded domain. We first introduce the artificial boundaries $Γ_±$ to get a finite computational domain. On the artificial boundaries $Γ_±$, we use the Laplace transform to construct the exact artificial boundary conditions (ABCs) to reduce the original problem to an initial-boundary value problem on a bounded domain. In addition, we propose a finite difference scheme based on the $\mathcal{L}1$_$2$ formule for the Caputo fractional derivative in time direction and the central difference scheme for the spatial directional derivative to solve the reduced problem. In order to reduce the effect of unsmoothness of the solution at the initial moment, we use a fine mesh and low-order interpolation to discretize the solution near $t = 0$. Finally, some numerical results show the efficiency and reliability of the ABCs and validate our theoretical results.