No it isn't. A hydrogen bomb is substantially more complex than a fission bomb, and miniaturizing it enough to be a missile payload is tough. The possible warhead that Kim was seen with is clearly a fission only device. A fission device is used as the trigger for an H-bomb, but there's a lot more to it than that.
NK will definitely test any thermonuclear device it develops rather than just assuming it'll work. It's something NK would love for propaganda purposes regardless of the necessity of the test. So, we'll know it when they have it - although the first version will likely be far too big for a missile warhead.
There is a lot of information in the open literature, which is worrisome. This diagram of the W88, our most advanced warhead and a 480 KT device, is on Wikipedia:
Having that diagram is still very far from having a working duplicate though... (BTW a similar diagram was on the front page of the New York Times in the late 90s when it was thought China received classified information on the W88. The Trident D-5 missile was designed to throw 12 W88s, although currently each missile carries eight to conform to START.)