Re: Intonation issue? Nut depth?
First, check that the bottom of your nut slot is about the same distance from the fretboard as the top of the first fret is. If it is slotted too low, this problem could result.
If it is the right depth, you can notch your nut back bit by bit in the area of the G string only. An angle is fine; the notch need not have parallel sides.
The shorter-than-normal scale of the Gibsons, coupled with the low tension of the G (usually the lowest tension string on the entire guitar), makes the G problematic on a Gibson (a Fender too, but more so on the Gibson), as the string doesn't bend far enough sharp when fretted in the lower register of the string.
All that being said, you should be judging intonation more based on commonly played chords than on single notes. What looks funky on a tuner, or even what sounds funky to the ear when played alone, might actually sound more in tune as part of a chord, i.e. how you might actually be most likely to use that note in real world playing.