I think you're having the exact same (really annoying) problem that just cropped up for me yesterday. First, let me apologize for my lack of brevity. I was a philosophy major in college, so I ramble..
I have a 2013 base Veloster (bought in Alabama---but not manufactured in the Alabama Hyundai plant, not that this matters). Had it for a few weeks, just over 1000 miles. Bluetooth worked flawlessly to begin with--which is good because that was one of the main reasons I got a newer car instead of an older model (ok, I also really loved the Veloster, but that's beside the point). Anyway...
Turned my car on after work and the bluetooth wouldn't work. Actually, when I push the "Phone" button, the phone screen popped up. It started uploading my recent calls, but when I pushed the green call button on the steering wheel, nothing happened. I went to the setup screen and tried to push "Bluetooth." The button lit up and then nothing happened. Annoyed, I pulled into a parking lot, turned the car off for a minute, turned it back on. The head unit reset, and I was able to access the Bluetooth settings and phone. For a minute or two. Then it stopped working again.
At that point, if I initiated a call on my cell phone, and had my phone (iPhone 4S with 6.1.2 iOS) connected to the Veloster, the call came up over the car speakers and the microphone worked, but nothing showed on the head unit. Eventually that stopped. And I still couldn't access the Bluetooth setup. Whenever I push the "Phone" button, it said "INITIALIZING BLUETOOTH HANDSFREE" up top on the screen for a few seconds, then went away.
When I got home, I turned the car off and back on a minute later. I was again able to access Bluetooth settings. I reset Bluetooth to factory settings (or whatever that option is) and it deleted all connected phones. Then I went through the process of re-pairing my phone. Everything seemed to work, though I didn't stick around for very long. I get in my car this morning, everything connects and I'm thinking I had fixed the issue. Nope. Worked for a couple of minutes, then I ran into the same exact issues as yesterday.
I actually tried to call Hyundai customer service yesterday evening, but the (very nice) lady didn't really know what she was doing. She was just trying to pull up the electronic version of the owner manual on her computer. I told her I could do that myself, so not to worry about it. Clearly this isn't a problem they have specifically identified and know how to trouble shoot. I guess I'll go to the dealer this weekend. I'm very annoyed because, as I said, Bluetooth was one of my favorite things about getting a new car. Oh, this problem also affects your ability to user Pandora (if that's something you do) since you have to go through Bluetooth to do it. Another thing I really loved about the car. I really hope they can get this (painlessly) fixed. Maybe just a software update. Because this, to me, clearly seems like a software issues, not a hardware (or user) issue.
If anyone has any idea how to fix it, I'm all ears. I wonder if that "ELA" user interface that can be installed via USB would fix the problem? I might try that if the dealer doesn't have any luck.