Is a one person (self-employed) defined benefit plan subject to required quarterly contributions?

Yes. However, the practical effect should be pretty meaningless because the minimum contribution is increased but it is fully deductible.

I thought the one person plan (where the participant is the 100% owner) was exempt from the quarterly contribution requirements because it was not covered by ERISA...or maybe this exemption only relates to the ERISA notice to employees about the failure to make quarterly contributions?

A DB plan is subject to 412 regardless of size, including 412(m). The notice you speak of extends far past just 1 person. I don't recall offhand how big a plan must be in terms of participants or assets for the missed quarterly notice to apply.