How do I record a vehicle trade in with a note and a trade in allowance?

Keep all the paperwork for your tax accountant, there is a form they fill out for this kind of thing that is different than the accounting entries you make.  You need a tax accountant for this one.

In QB

calculate and post partial year depreciation for the old vehicle.  If you have not been claiming annual depreciation for the old vehicle, stop here and see a tax accountant - the IRS does not care if you claim depreciation expense annually, but you must calculate what you should have claimed and apply it to this situation.

Create a new fixed asset account, for the new vehicle and and a new fixed asset sub account for  accumulated depreciation

Create a new long term liability account for the new note

Then journal entries ( *** means use the account balance for that account in the journal entry, and they need to done in this sequence)
debit accumulated depreciation old vehicle fixed asset account ***
credit old vehicle fixed asset account

debit new vehicle fixed asset account
credit old vehicle fixed asset account ***

If you paid a deposit, enter the payment and use the new vehicle fixed asset account as the expense for the payment

subtract the two notes, 44,234 - 20,658 = 23,576

debit old vehicle liability account, 20,568
credit new vehicle liability account, 20,568

debit new vehicle fixed asset account, 23,576
credit new vehicle liability account, 23,576