Is the sentence
- I heard her do that one day.
equivalent in meaning to the sentence
- I heard that she did that one day.
The first sentence sounds somewhat weird to me.
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No, they're not equivalent, and I agree that the first sentence sounds a little odd.
I would interpret the first sentence as "on one day in the past, I heard her do that", for example, if your former neighbour briefly tried playing the drums.
The second sentence is a bit better, and appears to mean that a third party told you that she did something. However, it's a bit ambiguous as to whether the "one day" refers to when she did it, or when you heard it.
I think you can probably say:
I heard she did that
I heard she used to do that
without any loss of meaning, unless there's some specific context I'm missing.