Ballentine's (3.49) and (3.50)

[andresB]

Those equation seems to not be derived anywhere but just stated without proof, are they postulated or one can prove them?

 

[ShayanJ]

Yes, they aren't derived from any other equation but they aren't postulated. They are natural, as the author explains.
The formula $e^{i v \cdot G}Ve^{-i v \cdot G}=V-vI$ is just derived in the same way as for Q and P. They are very similar, just here you have velocity space instead of ordinary space.
About the formula $[G_\alpha,Q_\beta]=0$, its just as simple as what the author says:

the position will be unaffected by the instantaneous transformation

Because velocity changes position only by the passage of time. So there will be no change at the instant of transformation. Also changing the position of the particle doesn't change its velocity directly.So the two operators are compatible.