1
general::
Phrase(s): on no account [and] not on any account
for no reason; absolutely not. • On no account will I lend you the money. • Will I say I’m sorry? Not on any account.
McGrawhill's American Idioms And Phrasal Verbs
2
general::
Also, not on any account. Under no circumstances, certainly not, as in On no account should you put a metal utensil in the microwave oven, or Dad said we can't go, not on any account. [Mid-1800s]
American Heritage Idioms