The fog cloud spell creates "heavily obscured" vision, which effectively acts like the blinded condition for characters in it, per the answer to this Q&A: Interaction between fog cloud and new vision errata rule
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: What happens to area-effect spells like Darkness or Fog Cloud when an Antimagic Field moves?
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?
Excellent Idea! (MAYBE there's a loophole)
First of all, I'm impressed by this strategy! It's a great idea, and turns the tables on the Beholder extremely well. You might have some trouble targeting the Beholder from within the cloud yourself, but characters normally know where other creatures are during combat (the fact that the Beholder hovers may complicate this, but probably not unless the Beholder takes the Hide action).
There is a small potential loophole I noticed, but it's not ironclad (more open to interpretation).
Legendary Action & Lair Actions
The Beholder can use its eye rays with its Action during its turn, but must choose targets it can see. However, it can also use its eye rays in two other ways: one is with a Legendary action at the end of another creature's turn. This legendary action is described as follows:
It can also use lair actions, one of which is the following (MM, p. 27, bold added):
Note that neither of these descriptions requires the Beholder to see its chosen targets.
It's likely that the definition of the eye rays given in the Actions section of its description is meant to apply to the other (Legendary) rays as well. But this is ambiguously worded, and open to interpretation. Clearly, some parts of the description of the eye rays must apply (or else there would be no limit to the range of these Eye Rays, for example): exactly how much applies, though, is ultimately up to the DM.
Even if its lair and Legendary actions do not require sight, note that your strategy would pretty much halve the amount of opportunities that the Beholder has to target creatures directly (as opposed to lifting a heavy object above a character while they are in the Antimagic Cone, and dropping it). This makes this strategy an excellent idea in any case.