Yes, there are ways. The easiest would be to use Trace, and ability that copies the opponent's ability. The text box will state what ability you trace, so it is a quick and easy way to find out.
If it is an ability like Water Absorb that has effect when a certain move hits it, you could be able to tell based on how the Pokemon reacts. You could also tell based on how the Pokemon acts; for example, you might be able to tell that a Pokemon has the ability Prankster if their status moves keeping going first when they shouldn't, or the game might say "Pokemon is exerting its pressure!" if the Pokemon has Pressure as an ability.
There is no button to press that will instantly reveal the Pokemon's ability, but by using logic and noticing things in battle, you can work out what the opponent's ability is with the right information.
