While it's not exactly ripe with details, the Hulu Plus for Android app's latest update promises increased playback resolution for three popular 1080p handsets: the Galaxy S4, DROID DNA, and HTC One. The assumption you might make, then, is that it's adding 1080p playback.
That's possible, though the amount of 1080p content Hulu currently provides is pretty limited, apparently. And Hulu itself lists a maximum resolution of 720p on its supported devices page. Last summer, the Hulu app was updated with "high resolution" support for small tablets, so my guess is rather that this new update merely allows 720p playback on phones, not the Full HD experience. Technically, it wouldn't really do much good to have 1080p video on a phone anyway - your eyes simply can't see that fine level of detail. Not to mention the significantly greater bandwidth and video processing loads of 1080p video.
Anyway, you can head to the Play Store to grab the update, which all address playback issues for certain LG devices, now.