For someone they might experience raised liver enzymes after 2 weeks, some 2 months, and some 2 years! It all depends on the person which is why I neglected to give any "official" timeline. But generally if you are going to take fluconazole for an extended period doctor's will usually check your liver periodically to make sure everything's ok.
If you are taking it weekly vs. daily though then yes it will have less of a potential effect on your liver. But of course you can't take fluconazole for the rest of your life, so your top priority should be getting your immune system back in check.