YouTube has all sorts of options for video quality playback when you visit the website, but in the Android app it wasn't until recently you were able to flip through and select video playback quality as an option, you were only given the ability to view in HD or not, and sometimes it didn't even look HD if you selected it. As of now any Android device that has the latest version of the YouTube app should be able to open up YouTube and play a video, then select either 480p, 720p, or 1080p as a video playback option. The good news for LG G3 owners, is that you should start seeing 1440p show up as a possible selection if you want even higher quality playback while you your favorite videos.
As far as we know this is limited to the LG G3 right now, as Droid Life pointed out that they had noticed the appearance of the option and we tested it ourselves with a G3 we have in hand. It does indeed show up as an option for video playback, so we have to wonder if this option will come to other devices that can support the playback quality in the near future. With that being said don't expect the 1440p option to come to devices that don't have a display that can support that playback quality, as we'll probably be stuck with 1080p playback and nothing more. That's not to say that 1080p playback on other devices that can't support the 1440p looks bad, because on the contrary 1080p playback on my Nexus 4 still looks amazing, but, it still isn't 1440p.
If you have an LG G3 open up YouTube and test it for yourself, since you'll have a device with a display that can support that video quality. If you see the option, than why not use it whenever it pops up. If you have never messed with the playback quality in the YouTube app before accessing the selection of different options is simple, you just tap on the video after you hit play and the settings button will appear, and hitting that will open up the panel that includes the different levels of video quality.