Will this breaker box accept a quadplex breaker?
This thing is a shut-off switch, basically.

Many air conditioning units have a shut-off switch near them.  This panel is designed to be that, and also provide overcurrent protection, and optionally GFCI.  It is a 2-space "panel" which is designed to fit exactly one 2-pole common breaker.  (any of the THQL THHQL TXQL or QPF types).  It is designed to provide a) a shut-off, b) overcurrent protection, and optionally c) GFCI to a single load.  It'd be a great hot-tub shut-off.   

GE's slick design with THQP and THQQP breakers uses a clever way to do double-stuff 2-pole breakers.  Since it uses half-width breakers, it simply uses a 2-pole breaker that sits in the halfway position, one contact on each bus bar.  Any number of 2-pole breakers can be stacked like this, if your panel is wide enough.  which yours isn't.  As such, the only supported 2-pole configuration with these breakers is the 2-pole THQP sitting in the middle, in the 2-3 position, with a half-width breaker on either side.  You cannot handle-tie the two outside 120V breakers because GE is quite specific that these breakers have an internal common trip mechanism.  Handle ties only assure common shut-off, not trip, so one side tripping could leave the other side energized - bad!

However GE doesn't seem to make a quadplex 2-pole for it.  That makes sense. The whole selling point of THQP is to avoid the duplex/quadplex model entirely. 

The bigger problem is your wiring.

This really nags at me:  Why would someone run 6 AWG to a shut-off switch to a single 30A load?  Makes no sense.  An installer would install 10 AWG throughout, unless he was some sort of visionary genius, and if he was, he'd have installed more than a 2-space "panel"!  So I deduce the wire back to the panel is probably 10AWG and cannot support a second circuit. You would need to replace it with 6 AWG to support a 23-30A main air conditioner and a 15-20A auxiliary load.  In fact, in some conditions 12AWG would be legal here!

Given that you may have to replace all that anyway... 

...you might go ahead and replace that tiny panel

Normally changing panels is a bugbear, but you've only got 2 cables coming in/out of it so far.  So now's the time. 

I recommend people get 2-3x the panel they think they need.  In your case since you need 2 240V circuits (4 spaces), I'd say go for a 12 or even 20 space panel.  Panel space is cheap, unlike regrets, and panels often come with free breakers!