A beholder's eye stalks (Monster Manual, p. 28) can target "one to three targets it can see within 120 feet of it." Thus like many spells, this seems to suggest it must see its target.
The central eye of the Beholder has an Antimagic Cone (essentially an antimagic field) in a 150 foot cone. Antimagic Fields simply temporarily suppress magical effects into which they come in contact. As the central eye turns away, the Antimagic Cone will sweep away in another direction and the fog cloud returns as suggested by the answer to this question.
If, however, the Beholder tries to use its eyestalk's spell effects to target something it can see in the cone, the Antimagic Cone prevents the magic from working.
Lastly, the Beholder has no dispel magic spells for the fog cloud.
In effect, with fog cloud, the Beholder either cannot see the characters to target them or when it can see the characters its eye stalk spells do not function.
Are we interpreting this correctly or is there some other way for the Beholder to use its spells in a fog cloud?