If I activate a creature ability that puts creatures on the battlefield, can my opponet counter it with a counter target creature spell?
No, you can't counter an ability with an effect that says
counter target spell. You can only counter a spell
One of the reasons why people play aether vial, because you can't counter creatures put into play with aether vial
You need to pay attention to the wording of the ability. Because if the ability says that you can "cast" the creature such as Garruk's Horde, than it does take the form of spell, and can therefore be countered.
But if the ability says to put it onto the battlefield than it's not a spell, and can't be countered