Note to Twelfth Night, 5.1.363, "Maria writ / The letter at Sir Toby's great importance"

This is not true. The writing of the letter was entirely Maria's idea (see 2.3.155 ff).

So, what's up? Maybe Fabian is lying, but it's unclear why he should be. Or maybe Shakespeare just forgot a fact of his own play because he was in a hurry to get to Hamlet.

BBC's Hamlet, the Closet Scene