To start a New Game in Myst on Oculus Quest, in the Main Menu, aim the laser pointer at the words New Game, then pull the controller's trigger. 
You may do this with either the left or right hand controller. 

Do not press the "A" button on the controller. 

Pressing the "A" button at that point (or any point) in the game will only serve to Reset Player Height. That is the only function the "A" button performs in Myst on Oculus Quest.