。
I have no intention of asking that person.
Literal
I [topic-は] that person [to-に] ask feeling [subject-は] not-exist.
気持ちはない literally 'feeling does not exist' — but here 気持ち carries the sense of 'inclination, desire, willingness.' The pattern [verb dict-form]+気持ちはない/がない means 'have no desire/inclination to [verb].' More personal and emotional than the cooler ~つもりはない.