How dangerous is indoor rock climbing with an inexperienced belay partner?

I'd agree with @Reustonium that Grigris will be safer initially, but it will be worthwhile learning with ATC's as when you progress to leading you will be thankful for the extra practice with them.

If your gym is anything like the indoor climbing centre I got my wife to learn at, the regulars will also be able to keep an eye out for obvious problems, such as tangling the rope tail, hands too close to the belay device etc. while I was up the top of a climb. They were also good at reassuring her while letting me down, as my weight was entirely in her hands at that point.

During the initial learning phase, you could also ask whether your belayer can route the rope through a ground level loop before it goes up to the top of the climb. Especially if one of you is much heavier than the other this can give you a bit more security against being pulled up into the air.

I wouldn't trust a belay partner who hasn't received basic instructions and has some practice. Using a device like a Grigri makes belaying very simple, but still has room for error. Belaying errors, regardless of the device used, can lead to a freefall.

For your first climb you should be sure to get instructions from a certified instructor. Once you're familiar with the basics of belaying you can also consider having a backup belay--it's a simple job that your instructor can train you on very quickly.

Provided both climber and belayer are correctly tied-in and using the belay device correctly then the single most important thing for a belayer to remember is to always have a firm grip on the brake end of the rope. They need to be sure to keep a grip on the brake end at all times that they have a climber on belay--and to remember to keep it there in the event of the unexpected. In gyms it's not uncommon for climbers on other ropes to swing very close to belayers, or when outdoors for dropped items or loose rocks to fall towards the belayer.