Filing instructions stated that I need to mail the W-2 and a form 1099-G that the government sends. I have the W-2 but never received a 1099-G. Can I mail without it?

Yes, you can mail it without the 1099-G form. Form 1099-G is only required when there was federal or state tax withheld. 

Since you didn't even receive yours, I am assuming it was for your state refund. 

If that not right and you do have tax withheld on your 1099-G, you will need to get a copy of the form to mail in.