They certainly could. Cicero in de legibus talks very lovingly about his home town Arpinum, going so far as to call it his "patria." His friend Atticus was confused about that - isn't Cicero a Roman? This leads Cicero to expound his theory of citizenship - every citizen has two fatherlands, one of birth, and one of citizenship. Each makes different demands on the citizen, but both deserve love from the citizen.

That's an imaginary conversation that happened in a philosophical dialogue, but you can find other similar sentiments elsewhere. Livy at the beginning of his history suggested that his hometown Padua might have been an imperial rival to Rome. Ovid called Sulmo his patria. The emperor Claudius was (in)famous for his patronage of Lugdunum, where he was born.

People could also adopt places as home. Atticus was famous for his affection for Attica - Cicero makes a point of it in de Leg. A Pollio is responsible for all kinds of infrastructure in Ephasus.