If the transition rate matrix X of one birth-death process is defined by
\begin{bmatrix} -\lambda & \mu \\ \mu & -\lambda \end{bmatrix}
and another transition rate matrix Y of a second birth-death process is
\begin{bmatrix} -\alpha & \beta \\ \beta & -\alpha \end{bmatrix}
And let $Z = X + Y$, is the transition rate matrix of $Z$ the following?
\begin{bmatrix} -\lambda - \alpha & \mu + \beta \\ \mu + \beta & -\lambda - \alpha \end{bmatrix}
In birth-death processes, it is assumed that arrival times are Poisson and death times are exponential. Since the sum of two Poisson processes is another with $\lambda = \lambda_1 + \lambda_2$ we can safely say the arrival time of the new process is going to be Poisson with $\mu+ \beta$ in your question.
The departure time (death) is the minimum of the 2 departure times. Departure times are exponential and minimum of 2 exponential is also exponential with the sum of underlying two rates. Hence, you can say the departure time is $-\lambda -\alpha$ in your example.
Combining the two, we can say that the new added process is another birth/death with ($\mu+ \beta$, $-\lambda -\alpha$), so the transition rate matrix you gave is correct.