It's the opposite, usually. "My heart dropped" is a sense of doom, foreboding, a realisation of oncoming cataclysm. Like 'as I saw the second plane come into view my heart dropped'.
Of course, people don't use language in a consistent way, and people will use terms thinking they mean e.g. their antonym.
It's probably a common initial reaction to learning of impending parenthood. Life will never be the same. Initially one might only see the looming challenge of the mountain to climb.
