Does proliferate add a poison counter to a player who already has at least one poison counter?

Proliferate (CR 701.27) lets you choose any number of players and/or permanents that already have at least one counter on them, then put one more counter of each kind already on them onto each chosen target. Because poison counters are counters, a player with one or more poison counters is a legal target.

The key restriction is the word already: a player must have at least one poison counter before you can proliferate to give them another. You cannot use proliferate to give a player their first poison counter — that must come from another source, such as a creature with infect or wither dealing damage.

Once a player has been given their first poison counter through normal gameplay, every subsequent proliferate effect you resolve can add one more poison counter to that player (assuming you choose them). This can quickly accelerate the race to 10 poison counters, which causes that player to lose the game (CR 104.3d).

Example: Your opponent has 2 poison counters from a previous attack by a creature with infect. You cast Thrummingbird and attack unblocked, triggering proliferate. You choose your opponent; they now have 3 poison counters. Later you cast Infectious Inquiry, proliferating again — they now have 4. Each proliferate added exactly one poison counter because they already had at least one.

Note that on your turn you choose which players and permanents to proliferate onto; you are never forced to target a specific player. If multiple players have poison counters, you may choose any combination of them (CR 701.27b).