The Super Lig leaders recorded a 5-1 victory at Rizespor earlier in the day, before making their way down the coast to Trabzon where the attack took place.
Several shots were fired at the front of the club's bus and the driver was subsequently rushed to hospital with facial injuries, though he appeared conscious in photos and videos posted online.
Club official Ilhan Eksioglu was left disgusted by the attack, wishing a speedy recovery to the driver before relaying his horror at such an "atrocity".
He told the club's official website: "Firstly, our prayers are with the driver. God has protected us all.
"There is nothing to say; this is an atrocity. Shame on those who encourage it.
"This is brutality. We are talking about guns, weapons. This outrage is before everyone.
"There is nothing else to say."
Fenerbahce were en route to Trabzon airport, ahead of boarding a flight back to Istanbul.
