To find the secret exit in Doom 64 map 18, Spawned Fear,
which leads to map 31, In the Void:

Not far from this level's entry point is a locked door that
requires a yellow key to open.
Once you have the yellow key, go stand next to that closed
yellow-keyed door, and turn to face it.
To the right of the yellow-keyed door there is a chamber.
A doorway to this chamber is a couple of paces around to
your right.
At the back wall of that chamber, on the left, there is a
secret panel controlled by a floor trigger and a timer.
This panel covers the secret-exit teleporter.

Open that yellow-keyed door and step through, down a very
short flight of steps.  If you continue on, around the
corner to your left, you'll come to another short flight of
steps.  Walk up those steps, and you might hear the secret
panel opening back in that chamber around the corner from
the yellow-keyed door.  If you wait a moment, you may hear
it close again.

Here's how to access the secret exit:

While the yellow-keyed door is open, Run across that second
flight of steps (to trigger the secret panel), and quickly
Run back up the first flight of steps, out the yellow-keyed
door and turn sharply around the corner to your left.  Look
for a new opening in the back wall of the chamber (towards
the left side), and Run through it.

The secret panel only remains open for a few seconds; If
you don't make it in time, simply go back to the steps and
try again.