While playing scrabble, the word mount was on the board, my opponent added ing, making the word mounting, then wanted to make a word off the g in the same turn, is that a legal play?
No, that is not a legal move.
From the official rules:
All letters played on a turn must be placed in one row across or down the board, to form at least one complete word. If, at the same time, they touch others letters in adjacent rows, those must also form complete words, crossword fashion, with all such letters. The player gets full credit for all words formed or modified on his or her turn.
Note that it says or - you cannot place letters in both directions.
So on the player's turn they could have added the "ing" to make "mounting" - but that would have been their turn.
If they also had a letter s for example, they could have played "sing" with the s adjacent to the end of "mount" to form "mounts", which would have legally created a new word and extended an existing one.