"Peeving about grammar disguised as a question"

Solution 1:

In fact this usage of the verb peeve, which is chiefly American, is old and well documented in the Oxford English Dictionary:

"...to grumble, complain petulantly..."

1912 Ade Knocking Neighbors 10 The Waiter peeved at being slipped a paltry $1.60.

Solution 2:

Whatever the logic, 'peeving' is a noun for 'the act of complaining'. 'Annoying' is simply an adjecting that the modified object annoys something else.

The verb 'to peeve' is rare: 'X peeves Y' sounds strange. The most common usage of the root is as a noun: X is a peeve of Y' translates to 'X annoys Y'. And the noun 'peeving' is most likely a formation directly from that noun rather through the verb form.