... See the Contact Organization at DigSafely.com. Be sure you tell them you're having an Invisible Fence installed around your property. If not, they may only 'mark' your lot near the house, or along 1 or 2 lines of your property. In short, you want to 'see' the markings for your ENTIRE lot.
They should give you a brochure describing, among other things, how to read the markings and how deep your existing utility lines are. That big green electrical box is probably a transformer which may provide service to a FEW other homes. Soooo, don't do anything until you call your Local One-Call System. It's free and they usually come out in 1-3 days.
If you're asking about the BIG green box 'interfering' with your fence, I'm sure it's possible. Have the box TOTALLY identified so you can ctc the mfgr. to see how far away from the box you should lay the line. Perhaps you can call the mfgr. of the fence to find out what info they need to make an 'informed decision'. THEN, when the One-Call Folks show up, you can ask them what the BIG green box is 'all about'.