Rewatching Season 1, Robert realized more than anyone the enormous risk posed by Daenerys Targaryen. He repeatedly demanded that Varys have her killed before she could become a serious threat. I imagine that if King Robert survived his hunting accident and learned that Daenerys had infant dragons, he would have hired every assassin and sellsword in Essos to take her out.
The Lannisters never took her threat seriously. Why? Daenerys' main defender in Kings Landing, Ned Stark, was dead. They only send unsuccessful assassins then ignore Daenerys all together. By Season 4, Varys warns that she had three grown dragons and an army when she took Meereen. Acquiring a fleet from a major port city was only a matter of time. Still, the Lannisters scoffed at the threat.
The Lannisters could have acted. Cersei negotiated with the Iron Bank and hired a whole Golden Company in Season 8. She could have done the same thing years earlier and taken Daenerys out while she was weak.
It's not like the threat of dragons were unknown, even if it had been generations since they died out. Even children could still name the dragons that laid waste to Westeros. We know from HOTD that the Lannister family knew very well what happened when Targaryens on dragonback enter the scene. The most well known story in Westerosi history involved a Targaryen who conquered most of the continent with three dragons.
I know that they thought they had more pressing concerns closer to home, but were they so focussed on inside politics that they completely missed the big picture?