Sometimes strings--and especially the G string on a Strat-style guitar--will have this problem. What it can be is that the string doesn't have quite enough angle over the nut. So when it goes from the nut to the tuning peg it's not at a sharp enough angle and therefore the string is kinda dampened at the nut. This usually is combined with loose tuners or something and causes the string to go out of tune often. You can check if this is the case by pushing the string down behind the nut on the G string while you play it. The note will go a bit sharp, but what you want to listen for is whether or not it sounds clearer. If it does then you've solved the problem. You can fix it by getting staggered tuning pegs or a string tree.

The way to do your intonation is to compare the 12th fret harmonic with the fretted 12th fret note. The idea is that the string should be exactly at its halfway mark at that point.