They had a conflict, sure, but by now means a major one. They made mention of other conflicts as well, like with the Zenkethi.

That doesn’t mean their society was all about war and conflict, it means they had border disputes. Conflict with smaller groups, like the Cardassians and Zenkethi, would not have nearly the effect as one with a much larger, much more powerful foe like the Klingons, Romulans, or eventually the Dominion (as shown in DS9).

With the latter, they have to specifically dedicate their resources conflict and war. With the former its mostly peacekeeping. Making sure their colonies and allies are defended while still being able to dedicate the majority of their resources to exploration and diplomacy. They won’t simply overrun the Cardassians, or the Zenkethi, as they’re likely potentially able to do (as im sure is implied with the Terrans in the mirror universe), as that is simply not part of their ideal.

Conflict with the Cardassians, or other smaller powers, is simply the price of being a large power. Conflict with a power more matched to their resource level, like Klingons or Romulans, would have more if an effect. Thus what you see in Yesterday’s Enterprise.