What is the value of a 1934 G US 10 dollar bill?

There was never a 1934 G US $10 bill. The highest series letter, which appears by the date, was D. The "G" is almost certainly the letter code for the Chicago Federal Reserve District that distributed the bill; that's not the same as the series letter.

Check to see what letter is next to the date. It could be blank or A to D. Then see "What is the value of a 1934 [letter] US 10 dollar bill?" for more information.