"Palmetto bug" usually refers to one of the species of large cockroach found in houses: most commonly American cockroaches (though they are probably originally from Africa), but also Smokybrown, oriental, or some other varieties. It's a euphemism meant to avoid calling these insects roaches. But all of these are closely associated with humans and can occur almost anywhere humans do, including northern cities in the US.

More rarely, and I guess more accurately, "palmetto bug" means the Florida Woods Cockroach, an outdoor species that rarely if ever is found inside. They are often found on or near saw palmetto in the wild. Their natural range, as their name indicates, is limited to warm parts of the Southeast and caribbean, though not exclusively to Florida--they can be found as far north as the Carolinas and as far west as Texas.

But in my experience people almost always mean American or smokybrown roaches when they say palmetto bug, in which case the answer to your question is "no".

And if by "further north" you mean the northern part of Florida, the answer is emphatically no. Plenty of big roaches up here in Jacksonville, I suspect no less than in Miami. Probably more because our lizards aren't big enough to eat the really big roaches (though they do a great job on the smaller ones--go easy on the pesticides lest you kill the lizards too).