Pliny the Elder in Natural History seems well aware of bird migration in the first century AD. He claims cranes fly from the east of India:

The tracts over which they travel must be immense, if we only consider that they come all the way from the Eastern Sea.

But he doesn’t know where storks come from:

Up to the present time it has not been ascertained from what place the storks come, or whither they go when they leave us. There can be no doubt but that, like the cranes, they come from a very great distance, the cranes being our winter, the storks our summer, guests.

He thinks swallows don’t migrate that far:

The swallow, the only bird that is carnivorous among those which have not hooked talons, takes its departure also during the winter months; but it only goes to neighbouring countries, seeking sunny retreats there on the mountain sides; sometimes they have been found in such spots bare and quite unfledged.

…and claims the thrush winters in the north, not the south:

they are often to be seen in places where they seek their food during the winter: hence it is that in winter, more especially, the thrush is so often to be seen in Germany.

Pliny also points out that geese and swans migrate, and interestingly even knows the V-formation makes for easier flying.

The flocks, forming a point, move along with great impetus, much, indeed, after the manner of our Liburnian beaked galleys; and it is by doing so that they are enabled to cleave the air more easily than if they presented to it a broad front. The flight gradually enlarges in the rear, much in the form of a wedge, presenting a vast surface to the breeze, as it impels them onward; those that follow place their necks on those that go before, while the leading birds, as they become weary, fall to the rear.

Pliny actually categorises some birds on their migration, or non-migration, which you can see in Book X.