By “intellectual conscience,” Nietzsche means the idea that it is wrong to believe something unless you have good reason to think that the belief is true. Someone with an intellectual conscience will form and endorse beliefs by applying the best epistemic standards known to him or her.
Most of us, Nietzsche thinks, lack an intellectual conscience:
[T]he great majority of people lacks an intellectual conscience. […] I mean: the great majority of people does not consider it contemptible to believe this or that and to live accordingly, without first having given themselves an account of the final and most certain reasons pro and con, and without even troubling themselves about such reasons afterward…. (The Gay Science §2.)
Exercising the intellectual conscience, if one does possess it, doesn’t necessarily lead to certainty; on the contrary. The more you subject your beliefs to scrutiny, the less certain you will become about them. This is good, because searching for the truth, as well as finding it, is good. Bernard Reginster puts the point as follows:
[T]he seeker after knowledge must want both knowledge and uncertainty or ignorance. He cannot be a genuine seeker after truth unless he actually wants to find it, but since what he cares about is the search after truth, he must also welcome the uncertainty and ignorance that supply opportunities for it. (The Affirmation of Life: Nietzsche on Overcoming Nihilism.)
Although intellectual conscience is a necessary condition, it is not, Nietzsche seems to think, a sufficient one. The “last idealists of knowledge in whom alone the intellectual conscience dwells today,” namely the scientists, have rendered Christian belief untenable, but they shrink from questioning their own values. “These are by no means free spirits: for they still believe in [the value of] truth.” (On the Genealogy of Morals, III §24.)
Properly employed by an authentic free spirit, intellectual conscience works against the tendency to assimilate novel experiences or problems to already-familiar explanations and evaluations. By eroding convictions and multiplying possibilities, it counteracts the disposition to simplify and unify and makes it less likely to get stuck in any one perspective.
The best article I know of on Nietzsche’s concept of intellectual conscience is Jeremy Page’s “Nietzsche on Honesty” (2019).
Below, William Holman Hunt, The Awakening Conscience, 1853. (Not a depiction of intellectual conscience, but it’s the best I can do.)