  In this paper, we study Bergman projection $\mathbb{P}_{\alpha,\beta}$ and
Toeplitz operators $T^{\alpha,\beta}_\varphi$ on the $\beta$-modified Bergman
space $\mathcal{A}_{\alpha,\beta}^p$. We give some properties of
$\mathbb{P}_{\alpha,\beta}$ and a necessary and sufficient condition for
$T^{\alpha,\beta}_\varphi$ to be compact. We end with a characterization of
Toeplitz operators via Carleson measures by introducing a new Bergman metric
inherited by the Bergman kernel $\mathbb{K}_{\alpha,\beta}$ that will be
equivalent to the classical Bergman-Poincar\'e metric.
