In Grimmet and Stirzaker, on page 258 it explains how to find transition probabilities, given a generator matrix:
(a) nothing happens during $(t,t+h)$ with probability $1+g_{ii}*h+o(h)$
(b) the chain jumps to state $j(\ne i)$ with probability $g_{ij}*h+o(h)$
Clearly the above is a function of the incremental time into the future, $h$. Then, observing equation (8) a little bit below on the same page 258, we can see that by taking the limit as h goes to 0 of $\frac{1}{h}(P_h – I)=G$. That makes sense for the above (a) and (b).
Now look at the example (15) on page 260: The generator matrix is $$ \begin{bmatrix} -\alpha & \alpha \\ \beta & -\beta \end{bmatrix} $$
And, solving for the transition probabilities (as shown on Page 267 and 268) we get the following: $$P_t= \begin{bmatrix} \frac{\beta+\alpha*e^{-(\alpha+\beta)*t}}{\beta+\alpha} & \frac{\alpha-\alpha*e^{-(\alpha+\beta)*t}}{\beta+\alpha} \\ \frac{\beta-\beta*e^{-(\alpha+\beta)*t}}{\beta+\alpha} & \frac{\alpha+\beta*e^{-(\alpha+\beta)*t}}{\beta+\alpha} \end{bmatrix} $$ using (a) and (b) on page 258, I would understand the transition probabilities to be
$p_{00}=1+g_{ii}*h+o(h) = 1-\alpha *h+o(h)$
$p_{01}=g_{ij}*h+o(h) = \beta*h+o(h)$
$p_{11}=1+g_{ii}*h+o(h) = 1-\beta *h+o(h)$
$p_{10}=g_{ij}*h+o(h) = \alpha*h+o(h)$
Which set of transition probabilities is correct?