The probability of an event $T$ is given by:
$$ P_{T} = \frac{\beta^{\alpha}}{\Gamma(\alpha)}\displaystyle\int_{\gamma}^{\infty} T^{\alpha-1}e^{\beta T}dT$$
Given that I know $P_{T}$ and $\alpha \in \mathbb{R}_{++}$, $\beta \in \mathbb{R}_{++}$, how can I solve for $\gamma$?
I know if $T$ is Gaussian distributed, it is possible. But what if $T$ is Gamma distributed as it is here?
As written in the post, the integration requires that $$\Re(\beta )<0\land (\gamma >0\lor \gamma \notin \mathbb{R})$$and the problem finally reduces to finding the zero of$\gamma$ $$f(\gamma)={\Gamma (\alpha ,\beta \gamma )}-P_T\,{\Gamma (\alpha )}$$ which requires numerical methods (just as angryavian already commented).
From a practical point of view and for numerical efficiency, it would probably be better to find the zero of $$g(\gamma)=\log\left({\Gamma (\alpha ,\beta \gamma )} \right)-\log\left(P_T\,{\Gamma (\alpha )} \right)$$ which is more linear (making Newton happier).