In his Description of Greece, Pausanias comes to the topic of the age of Homer and Hesiod and begs off discussing it, though he admits giving much thought to it, because of the character of people who work on such things:

“It would not be sweet for me to write about the relative age of Homer and Hesiod, even though I have worked on the problem as closely as possible. This is because I am familiar with the fault-finding character of others and not the least of those who dominate the study of epic poetry in my time.”

περὶ δὲ ῾Ησιόδου τε ἡλικίας καὶ ῾Ομήρου πολυπραγμονήσαντι ἐς τὸ ἀκριβέστατον οὔ μοι γράφειν ἡδὺ ἦν, ἐπισταμένῳ τὸ φιλαίτιον ἄλλων τε καὶ οὐχ ἥκιστα ὅσοι κατ’ ἐμὲ ἐπὶ ποιήσει τῶν ἐπῶν καθεστήκεσαν.

(The more things change….)

According to the biographer of sophists, Diogenes Laertius, the 4th century Heraclides Ponticus wrote “Two books about the age of Homer and Hesiod” and “Two books about Archilochus and Homer” (Περὶ τῆς ῾Ομήρου καὶ ῾Ησιόδου ἡλικίας α′ β′, Περὶ ᾿Αρχιλόχου καὶ ῾Ομήρου α′ β′; see Koning 2010, 40).

Of course, antiquity presented every possible opinion on this:

Suda s.v. ῾Ησίοδος

“He was according to some older than Homer; but according to others he was the same age. Porphyry and most others argue that he is one hundred years younger…”

ἦν δὲ ῾Ομήρου κατά τινας πρεσβύτερος, κατὰ δὲ ἄλλους σύγχρονος· Πορφύριος καὶ
ἄλλοι πλεῖστοι νεώτερον ἑκατὸν ἐνιαυτοῖς ὁρίζουσιν.

(You can probably expect more of this. Palaiophron and I are developing a fixtion with the silliness of the Suda. He has confessed to fantasizing and developing “a book of refutations titled Suda Says: Everything that Classicists Know is Wrong.”)

Like this kind of stuff? These books are good:

Barbara Graziosi. The Invention of Homer. Cambridge, 2002.

Hugo Koning. Hesiod: The Other Poet. Leiden, 2010.