We might!

It's very possible that sharp-eyed observers could have gained clues that the Azores were out there long before their official discovery; many migratory bird species stop there, and anyone watching birds moving northeast from the Azores could have deduced they were coming from islands to the southwest.

There is a tale from the 18th century that a hoard of Carthaginian coins displayed in Portugal was discovered in the Azores, but the supposed site was never charted and the coins themselves lost.

Mice in the Azores are largely descended from ancestors in Iberia; however, mice on some islands have DNA from ancestors in Scandinavia. This hints that Norse explorers may have visited the islands centuries before the Portuguese.

Many medieval maps contain islands in the general vicinity of the Azores, but there is very little consensus on whether this reflects actual knowledge (passed down from Norse explorations or Iberian fishermen who guarded the location of their most lucrative fishing spots and secluded beaches) or inflated legends which happened to match up with the location.

Most intriguingly, there may be a series of hypogea, or underground tombs, on the islands. A team led by Nuno Ribiero excavated there about ten years back and claimed they'd found several sites in natural caves, including Paleolithic rock art. As far as I can tell, though, these claims either haven't stood up to closer examination or are at the very least too preliminary to confirm the findings.