Here's the video to see how I did it.If you prefer written: When the game gives you the chance for going outside of the HQ for the first time, you can pick this quest up from Lili. She is at the right side of the The Quarry when you leave the HQ.
I'd recommend picking this when you're first got outside and don't try to do it
until you got the Time Bubble PSI power. When you get the power, go to the Questionable Area? waterfall and hop on to the
middle log before it goes up. Just before it goes up, use your Time Bubble power. It'd be way easier if you have less cooldown between Time Bubble upgrade too, so you can spam it. Make sure you're facing towards to waterfall to see where you need to jump, there'll be a gap. Jump there and grab that mushroom and go back to Lili.