Motorola never made a phone specifically aimed at gaming, but if they did, it would probably have looked a lot like this concept from 2011 that recently made its way onto the internet, courtesy of the Behance community's Ye Xu. Xu purports to have actually collaborated with Sung-Kyun Bai, the lead designer of Motorola's Atrix 2, on this concept. It features buttons arranged in a way that would be conducive to use as gaming shoulder buttons, along with stereo speakers, and a retractable second front camera.
The back camera on this gaming phone is situated among the speakers, and gives the phone a Droid RAZR-esque bump, but along the length of the phone rather than across it. The screen looks fairly big, likely on par with modern 5-inch devices and others around that range. A fingerprint scanner is also on the rear. Interestingly, the retracting camera looks like it could rotate, which could give the phone 3D camera capabilities. This could also mean 3D audio, given the speaker setup. Moving to the internals, considering Moto's lineup around that time, this phone would likely have boasted a 1,280 x 720 pixel screen, along with an NVIDIA Tegra 2 or Tegra 3 processor, depending on the release date. For the time, 1GB of RAM was generous, so this would have probably been the case. 2GB of RAM didn't become the norm for flagship devices until 2012's Samsung Galaxy S3 popularized it.
This phone would have come out around the time that Moto was busy innovating and diversifying their lineup, with phones like the high-powered Atrix and its unique Lapdock accessory, and the strikingly beautiful Moto Droid RAZR. This, of course, means that it would have fit right in with Moto's lineup at the time. Ye Xu doesn't say exactly why the concept never saw the light of day, but given that it was conceptualized in 2011, it may have been bad timing. A TI OMAP processor wouldn't have done very well for gaming, and NVIDIA's Tegra 3 wasn't going to come out until near the end of 2012, while the Tegra 2 was showing up in normal consumer devices, meaning that it likely wouldn't have had the grunt to justify being put into a specially made gaming phone.