According to this comment from the general question Is it possible to create a quine in every turing-complete language? it seems like it is said that it's possible.
However I didn't find any Ook! Quine on the internet.
Do you think that it's really possible?
And if yes will we be able to find it?
It wouldn't even be very difficult. You would want to code it in brainfuck and then translate, and the internal representation for each command should be a pair of numbers (probably from 0-2) to represent the punctuation of each half-command. You could borrow much of the structure from Erik Bosman's brainfuck quine.
Updated: here. https://gist.github.com/danielcristofani/1fe53487df1f7afcb5b91c06d95184b2
This is ~40 commands taken directly from Erik Bosman's quine, another ~120 freshly written commands of rather clunky output code to handle Ook!'s verbosity, and then the data segment to represent all that.