You have a car. You have to go to the same way as Steve, who is on foot. You offer him a lift. You say to him:
The exercise is from Can/Could/Would you ... ? etc(Request, offers, permission and invitations) unit. The correct answer is obviously Can a give you a lift? but I never heard people saying this.
I would use something like this
Hey, I have to go the same place as you, so I can give you a lift if you want.
Hey, I have to go the same place as you. Would you like me to give you a lift?
I'm just curious if Can I give you a lift? is a natural way to say this or it's grammatically correct, but not used by native speakers.