More likely the nut is too HIGH for the thicker G string, if the problem is worst over the first few frets.

Alternatively the nut for the fat G could be chipped so that in effect the nut for that one string is too far back. If so a temporary cure can be effected with superglue and masking tape.

If however the problem gets steadily worse as you go up the neck, it MAY be that string has gone conical and a new fat G string might cure it. More likely it is the classic 12-string intonation problem that each course of octave strings really needs "compensation" (like b-compensated saddles do for the b-string) at the saddle. I am told this can be done but I have never seen it. Could your fat G have become mis-seated at the saddle?