An odd request, I know.
What I'm looking for is a formula on the Account object that grabs the most recent Contact name that is assigned to an opportunity / quote.
Is this possible with a formula or should I be creating a flow to accomplish this?
It's not possible to use a formula by itself to do this because there is a many-to-one relationship between Opportunities and Accounts and similarly, complex relationships between Contacts and Quotes. Formulas are not able to reference multiple children or multiple records that look up to the record with the formula on it.
You will need to use a flow or an Apex trigger to grab this data when the associated record is updated.