I think (hope?) they meant something like "She will offer to forget about this." As in, graciously turning a blind eye to something potentially embarrassing but not dangerous... I'm reporting it.
This is really clumsy English. Instead of using the gerundive "forgetting about," you should use the infinitive here: "to forget about."
It is definitely wrong "לשכוח זה"
You need at least the word את
But "לשכוח את זה" means "forget it". You lose the about.
The answer is right. "לשכוח מ" is a phrase that is roughly equivalent to the English "forget about"
Come to think of it, the Hebrew is ambiguous. Is she suggesting that you forget about it, or is she suggesting that she forget about it.
So IMO the translation could be "She will suggest to forget about it", but it could be "She will offer to forget about it"