*sigh* I wish the people who wrote these examples would pick a verb tense and stick with it. On the other hand, if they did that, I wouldn’t have any examples dealing with tenses…
Change lifted to lifts.
Delete quietly, not because it’s an adverb but because it’s an unnecessary adverb. Whispers are always quiet, unless otherwise specified (as in a stage whisper, which is actually rather loud so it’ll carry), so there’s no need to say quietly whispers.
The plane lifts off the runway and into the air. The person next to you turns and whispers in your ear, “I know I’m supposed to keep this a secret, but I absolutely must tell someone.”
If you want to emphasize that the dialogue happens after the plane takes off, try this: After the plane lifts off the runway and into the air, the person next to you turns[…] This way you’re maintaining consistent verb tenses but still making sequence extra-clear. (Do not write After lifting off the runway and into the air, the person next to you turns[…] You can see why that’s wrong, can’t you?)