Before answering that you do obviously understand something out of false sentences, too, consider that this would lead to:

—distinguishing between understanding the meaning of a sentence and knowing it to be true

—assuming a non-committal understanding of the meaning of a sentence

—understanding fitness as a requirement for the sentence meaning (yogyatā) as limited to the lack of obvious inconsistencies and not as regarding truth

—(possibly) assuming that the meaning of a sentence is not an entity out there (since there is no out-there entity in the case of false sentences), but rather a mental one

If you are now inclined to say that Indian authors on a whole could not answer yes to the question in the title, read the following sentence by Veṅkaṭanātha:

śaśaviṣāṇavākyād api bodho jāyata eva

Also out of the sentence claiming that hares have horns (e.g., out of an obviously false sentence), an awareness does indeed arise (SM ad 1.1.25, 1971 edition p. 114).