Even in peace time I highly doubt that
I rather think a centurion with 80 soldiers and 20 noncombattant under his command would have better things to do.
He would rather give the order to one of the 10 conterbiniums or even half such a unit to fix a criminal to the cross.
Or even more likely, get a thrust-worthy veteran roman soldier and assign some auxilairy troops to him to get the job done.
In roman cities there would be no legionairy soldiers stationed normally.
That doesn't mean some enterprising retired soldier couldn't crucify someone with some help of slaves/ hired people to crucify a criminal.
Specialized centuries would be out of the question.. for example:
The whole of Judea , Sameria and Gallilee wouldn't have more then 4 centuries of legionairs in peace time during beginning of AD.
The bulk of the closest legion was based out of Syria where the desert was less severe and frontlines with parthia were much closer to the roman lands.
Legionairs were combat troops... the elite troops if you please.
They need to be kept in optimal fighting condition... marches and weapon training.
In a full legion yeah maybe they had some specialists who performed crucifixions if needed in addition to other duties.
But crucifixion wan't standardized.. think most roman legionaires could perform a sort of crucifixion.
Nothing to fancy mind you
Secure forearms/wirst with ropes or nails. secure feet with ropes or nails.. the nearest tree will do nicely.
or any wooden structure for that matter..
If victim was in danger of ripping free of the nails.. just add more nails/ropes..