Almost nothing is known of his life, but it is likely he was a praetor before 218, for when Hannibal besieged Saguntum (now Sagunto), an ally of Rome, and the Saguntines petitioned Rome for assistance, the Senate sent Baebius and Publius Valerius Flaccus as envoys to Spain, with instructions to demand that Hannibal leave Saguntum alone, and then to proceed to Carthage to ask for his surrender as punishment for breaking the treaty that had been concluded at the end of the First Punic War. Turned back in Spain, the delegation got a hearing in the Carthaginian senate, but the Carthaginians supported Hannibal. (Livy 21.6)
Later that year, Baebius was again part of a delegation of five - described as "older men" by Livy - sent to Carthage to formally declare war, after which the envoys visited Spain and Gaul to recruit allies, with mixed results. (Livy 21.18)
After that we hear no more of Quintus Baebius, but his sons Gnaeus and Marcus both attained consulships.
See also: Baebius