A character I have lies on why they survived in the ocean by saying they were in a blue whale's mouth. Is it possible to survive in a blue whale's mouth if the blue whale is underwater? Half-hour duration at most. No equipment. Is there some trick for surviving the pressure change and getting air within the whale?
A blue whale's throat is only about 12 inches wide, so it cannot physically swallow you even if it wanted to. You could fit inside it's mouth but the blue whales do not eat anything bigger than krill and it would likely try to push you out of it's mouth with its tongue to prevent a throat blockage. The biggest threat is that you would be crushed or seriously injured by its massive jaws and tongue flexing and working around to try to force you back out, think about you trying to dig a popcorn kernel out of the back of your mouth with your tongue, in this situation you are the popcorn kernel, only you are a lot more squishy.
Even assuming you only sustain something like broken bones an injury underwater is extremely more dangerous than one on land for obvious reasons. You could theoretically survive but would in all likelihood not be in very good shape. Blue whales are so massive that they are capable of crushing a human by pure accident. There have been several accidents involving tourist whale watchers where blue whales capsized boats by accident, one man was hit by a blue whale's flipper as it rolled over and was thrown so hard against the control console of his small boat that he needed major reconstructive facial surgery. Blue whales are so big just rolling over and whopping you with a flipper is about like getting hit by a small car.
(assuming you are wearing SCUBA gear in all three of these scenarios)
Most likely scenario: You are crushed within the whale's mouth and die before being spit out from overwhelmingly catastrophic injuries and asphyxiation due to crushing.
Possible scenario: You are severely injured within the whale's mouth and spit out still alive, but are too badly injured to swim to the surface and drown.
Most unlikely (nearly impossible) scenario: You are taken into the whale's mouth while it is feeding with the large volume of water that it is filtering krill from. It continues on its way for a very short time (a few minutes) lazily filtering the water before it realizes something it doesn't want to eat is in there and spits you out, battered, terrified, but still in decent enough condition to swim away.
The oral cavity is a not a cavity when closed: the tongue fills it up.
So, if you happen to be in there, you will be squeezed between the tongue and the upper wall of the mouth. Not a comfortable place, and not a lot of air to breathe. Add the pressure of the tongue, and you can hardly survive more than few minutes, depending on your apnea skills.
Is there some trick for surviving the pressure change and getting air within the whale?
No, no trick, you just die.
If you were in some kind of hard suit, like an atmospheric pressure dive suit or a set of old school plate armour you'd actually have a better chance than you might think of surviving for a few minutes. In fact a lot of those diving suits have an air reserve so you could last for as long as that does. Without some form of crush protection though you're going to get squashed flat by the pressure when your whale dives.