What is a computation? According to Alan Turing (writing in 1936/7), it is this:
A computation occurs when the human mind carries out a mental action according to a rule.
Computability arises from a human mind carrying out rules.
What is an intuition? It depends on how the word is used and in which context it's being used. In Turing's case, we (or a mathematician) use our intuition when seeing the truth of a formally unprovable Gödel sentence. Gödel sentences can't be proved. Nonetheless, they're true and they're taken to be true.
Another way of looking at intuition is with another of Turing's ideas: the “oracle”.
In the case of a Gödel sentence, the mathematician (or the oracle) simply “has an idea” that the Gödel sentence is true. That is, he doesn't use a mechanical method to establish its truth. He has an idea (or an intuition) that it's true.
You may now ask how something can be established as true - especially in maths - without proof. You may also ask how truths can be established - especially in maths - only on the flimsy basis of a mathematician's (or even on hundreds of mathematicians') intuition or his simply “having an idea”.
Computer scientists - and the philosophers of mind who focus on computer/brain comparisons (or who even see the brain-mind as a literal computer) - will like Turing's conclusion (of 1945) that algorithms are enough to account for all mental activity. Bearing in mind the previous comments about intuition, Turing thought that algorithms also encompassed non-mechanical intuition.
Just as intuition followed algorithms (therefore rules), Turing believed that “initiative” didn't require uncomputable steps. In other words, human and computer initiative is also a mechanical process. (That would make the idea of computer's showing initiative - or intuition - less problematic for the simple reason that what it's doing is still a computable or a mechanical process.) However, as stated earlier when I mentioned the fact that computers may go beyond the rules (or programmes) laid down by the programmer, even if a computer departs from the computations which were programmed by the programmer, it would still be following a (new) rule, indulging in computations, or following mechanical processes. Indeed what else could it be doing?
Another way of looking at a computer's - or a Turing machine's - ability to follow its own rules (or to show initiative) is for the programmer to engineer an element of randomness into the computer (or into the programme). That was what Turing did with his Manchester computer of 1948/50. That seems to mean that such randomness (as it were) brings about intuition or initiative. However, it would still be intuition or initiative that's grounded in computation (or rules/mechanical processes). The randomness, therefore, would simply be a result of the computer not abiding by the programmer's rules (or programmes). It doesn't mean that the computer has gone beyond rules or computations.