ZTL Controller reported lack of proper interphone coordination released an uncoordinated departure; resulted a MVA CFTT event.

Narrative

I was working the Radar position on combined Sectors XX/XY; it was busy so I had a XX-side and we were handling the sector well. The XX-side was offered a break and took it; and as the briefing was finishing [Approach] ZZZ East called on the XXX line. The new XX-side; Person A; answered the line and I continued to work the traffic. However; I could tell it seemed like [Approach] ZZZ wasn't responding to him; and eventually he hung up and verified that I had heard them call 'North Departure'; which I had; but when he had gotten on the line they were already in communication and had said they didn't need to talk to him anymore. He called back about 3 minutes later to cancel a block they had approved for us and verified that they didn't need anything from us; to which they clarified that the coordination had already been completed and there was nothing further needed. I didn't hear these calls myself but that was the information I received from Person A. A few minutes after that; we saw a departure off of ZZZ; climbing north bound; which we had not approved and nothing had been coordinated with us. We called the aircraft multiple times with no response; and finally Person A was able to get [Approach] ZZZ on the line and they said the aircraft had been approved direct ZZZZZ; or on a 360 heading which would be almost the exact same heading; climbing to 5;000 ft.; which whether on a heading or direct ZZZZZ was putting them in an ZZZ1 of 5;400 ft. We had never approved this; nor had we been made aware that anyone else had approved it. We were able to get radio communication with Aircraft X and verify that they could maintain their own terrain and obstruction clearance through 5;400 ft.; to which they responded affirmative; and I cleared them up to their cruising altitude of 7;000 ft.; on course leaving 5;400 ft. As I didn't answer the line and Person A had a hard time getting any info out of [Approach] ZZZ; this is all conjecture; but it seems clear another sector answered the XXX line; either didn't state their sector name; or they did and [Approach] ZZZ didn't realize it was the wrong sector; and then a departure was approved in our airspace by a controller who is not aware of any of the specifics of that airspace.This situation would have been avoided if proper procedures had been followed; the receiving sector announcing their sector name and the calling sector ensuring they are talking to the appropriate sector.