G is where the neck octave key closes and the body octave key opens.

It may be mechanical and not you. With your right hand, while playing high G with your left, make sure the neck is closing, push it down. Then make sure the body octave key is opening, by lightly lifting it if necessary. Look at what the keys are or are not doing. Do they move freely as you move from A to G with the octave key activated?

Sometimes debri may get into either hole and make things not free blowing.

Run a light into it and see if any pads have a leak.

Is the saxophone new or old?

If it is you, just keep at it.

Good Luck

The octave key problem described by historicsaxwhisperer is a very likely culprit.  You sound very new to the sax, so I am not sure if you are familiar with the two octave keys.  If you are, not, you might want to take the horn in to a technician and do 2 things:
1.  Play it and demonstrate the problem.
2.  Have the techician play it and see if he has any problems.

This may help determine if it is you or the sax.  Good luck.  If you get this issue resolved, please return to this post and let us know.