Adverse possession occurs when you occupy property exclusively (not shared) continuously for an extended period of time (e.g. 30 years).
Rent moratorium: NO.
The mechanism is simple: your rent is being paid for you by the government or whoever organized the rent moratorium, or your rent is being statutorially set to $0.
That means your occupancy is consensual and agreed, and due rent is being paid (even if $0), and it qualifies as normal rental activity. This resets the clock on adverse possession.
Eviction moratorium: No, because of that
What's happening right now in most jurisdictions is eviction moratoriums. That means You still owe your rent, however the landlord's right to resolve disputes by evicting people has been suspended because that would drive people into homelessness and shelters, and those are a hotbed of COVID-19. (that rationale is fading fast, since sheltering-in-place no longer has the importance it once did.)
Once, a criminal decided that since lemon juice makes invisible ink, rubbing lemon juice will make you invisible to security cameras. While being stuffed into the back of the police car, the crook exclaimed "But I used the juice!"
Similarly, a few people have decided that suspending evictions means they don't have to pay rent. That's going to turn into a complete disaster for them, but it is certainly a hostile, overt act, which controls the property exclusively, and the landlord knows it. So does "not paying rent" count toward adverse possession?
No. Adverse possession only works if you never had permission to be there, so it is excluded to tenants, even tenants who are late on rent. This means you must first end your tenancy by vacating the premises. Which certainly could happen during an eviction moratorium, simply by moving out.
And then, the tenant would have to somehow re-occupy the property, so they become squatters. And then, the landlord would have to release control of the property to the point where they were no longer paying property taxes or doing maintenance.
That's quite a long chain of events, and you would need a particular situation for it to work.