"Do you forgive me?"
Translation:Vergibst du mir?
“Me” can mean either mich or mir. The difference between these forms is their grammar case. Mir is DATIVE personal pronoun and mich is ACCUSATIVE personal pronoun. There are specific verbs that match with dative and verzeihen/vergeben is one of them! You can find an online list with this kind of verbs, they are called dative-verbs. Have a nice Evening
On this list http://germanforenglishspeakers.com/reference/dative-verbs/ there is no sign of vergeben . I checked other lists also. (verzeihen IS dative, however).
Perhaps because vergeben can take two objects: the direct object is an offence, the indirect object is an offender.
So for example Ich vergebe dir deinen Wutausbruch (I forgive you your fit of rage).
Most commonly, though, only the offender is named: Ich vergebe dir.
(And if the offence is named, then it's more commonly in a prepositional phrase with für, in my experience.)