English uses the phrase abuse of trust as well, but it would not be used in the kind of sentence you proposed.
Here are some examples of abuse of trust taken from English-language periodicals (compiled by The Free Dictionary)
"Your actions in pursuing an emotional and sexual relationship, and in having sexual intercourse with [a patient receiving psychiatric care from you], were a gross abuse of trust."
"And a new offence, abuse of trust, will stop adults exploiting under- 18s in schools, the forces and places of care."
"In a gross abuse of trust, Pringle, one of a small number of people with access to the committee room, began taking cashed tickets and re-cashing them for himself."
The last example is perhaps the easiest to explain. Pringle was given access to the committee room, where something valuable was stored, because people trusted him. They trusted him not to steal the valuables. He abused their trust by stealing the tickets.
Asking for a favor from someone may be an abuse of trust, but probably not in the case you described.
If you convince someone to let you borrow their car by lying to them, that may be an abuse of trust. If you repeatedly ask to borrow someone's car, but are always honest about the reason, there is no abuse of trust, but it may be an abuse of their goodwill.