Is the music system the server or receiver? The phone has Miracast so by default it has DLNA. Type dlna in a play store search and you will find dozens of apps. You just need to find one your comfortable with. I found HomeDia from Qiss to be the easiest to use. None of them come with very good instructions.
No, unfortunately, Moto X doesn't have DLNA. I just got mine yesterday and was trying to figure that out myself. That was a feature I didn't verify, assuming it would have it. It does work with Miracast, which mirrors your phone's screen on the TV screen. And, it does work with Chromecast for the few apps that support that now. (with tab cast feature from a desktop, you have a little extra ability, with sharing a browser window, and even (experimentally) your desktop, but only from a PC running Chrome browser, not mobile Chrome). But, as other posters mentioned, you will have to find a DLNA app on the Play Store for true "DLNA" uses, where you can wirelessly access your media like a harddrive and play them without relying on the phone except to serve the media files. There are a lot. Just check reviews. The device you have as your DLNA receiver (renderer) is important, too. I have an Xbox I use with my Droid Bionic, and it works still with DLNA apps with my Moto X. With both phones, DLNA is cludgy...often I have to reboot my phone and Xbox for the video to play smoothly, but then it does as long as I don't try to rewind. Good luck! I did read that (at least) 2 major Android devs are working on ways to use Chromecast to play local videos and videos from Google Drive. But, possibly won't ever be available for stock Android because of trying to protect DRM material from being mis-shared.
Oh, another thing...the Moto X does have WiFi Direct, which eventually may replace DLNA. I haven't looked for any apps to make use of that yet, except as part of a Miracast setup. Miracast only works for Wireless Display, though, not anything else (yet).