Various things, particularly the history of the illithid race, has changed throughout the various versions of D&D. Which of these are to be considered canon seems difficult to decide as there are many retcons as well.
On the other hand, amongst these conflicting sources, there is one which distinguishes itself in terms of its completeness. The Illithiad is an AD&D 2e sourcebook that specifically focuses on illithids. And in this sourcebook a detailed description is given according to which an adult illithid can only survive up to 4 months without consuming a sentient brain:
Biochemicals and hormones found only in brain tissue are essential nutrients for adult illithids. However, illithids eat brains more for the psychic quotient contained in sentient gray matter than the physical nutrients. ... Illithids use this liberated psychic essence to reenergize their metabolism so effectively that the consumption of only one brain a month constitutes a healthy illithid diet.
lllithids do occasionally snack on the brain tissue of animals that they consider culinary delights - such as ustilagors, the immature forms of intellect devourers. However, illithids become malnourished and weak in the absence of the minimum essential sentient brain per month. For every month an illithid goes "hungry," it loses 1/4 of its original hit points and Psionic Power Points, until it finally starves to death following the fourth month of privation.
It is also worth noting that rogue illithids will encounter other problems beyond nourishment. They will feel uncomfortable having lost contact with the elder brain. They are also likely to be hunted, as "traitors are executed immediately" by the community.