Yes, pressure treated lumber can be painted, as long as it is dry. I painted my mailbox post about 5 years ago, and it still looks great. Except for the dust and bird crap.
I am a big believer in priming any surface that has to be painted. Making sure the post was dry, I applied 2 coats of an exterior primer, then the finish coat. Before placing the post in the ground,I wrapped the part that went into the hole with a layer of #15felt paper.
Wood must be dry.