I don't think my father would willingly lend me the car.
Literal
Father [subject-が] gladly car [object-を] lend-for-me [quotative-と] [topic-は] think-not.
~とは思わない negates the embedded thought: 'I don't think that...' 快く means 'willingly/gladly.' 貸してくれる = 'lend for me' (benefactive). The は in とは adds slight emphasis to the negation.