The boundedness of the Toeplitz operators induced by positive Borel measure between weighted Bergman space $A^{p_1}(\omega_1)$ and $A^{p_2}(\omega_2)$ over the upper half plane are characterized, where $0<p_i<+\infty$ and $\frac{\omega_i(z)}{(\mathrm{Im} z)^{\alpha_i}}$ ($\alpha_i>-1$) are in the B$\grave{e}$koll$\grave{e}$-Bonami class $\mathcal{B}_{p_0} (\alpha_i)$ ($p_0>1$), $i=1,2$. Khinchine's inequality and atom decomposition techniques are used in the proof of the main result. This is a joint work with Changbao Pang and Yan Li.