Can a Triac be driven using only positive voltage due to the laws of Q1 and Q4 quadrants?
Yes, it's possible and should work OK but the whole circuit must be considered live.
Table 3 of the BTA12 datasheet states that for a &quot;Standard triac&quot; IGT is 25 to 50 mA for quadrants I - II - III and 50 to 100 mA for quadrant IV at VD = 12 V, RL = 30 Ω.
@Spehro says that the -600B is not rated for quadrant IV switching. I'd believe him but I can't see it in the datasheet.
A major concern is that you might be considering driving a mains triac directly from the microcontroller and that the it's ground will now be referenced to mains voltage and must be considered live. This in turn means that you must ensure that the whole device including USB port is isolated properly and you can't connect to a laptop, etc., for serial port debugging, etc.
There is really no need to skimp on safety. An opto-triac provides the required level of isolation for, probably, 50c or so. There are plenty of application notes for the MOC series of opto-triacs and plenty of sample circuits on this site.

Figure 1. A sample opto-triac control. Image source: Electronics Tutorials.
You'll find more information in the MOC 3020 datasheet.

Use the MOC 3020 if you want to be able to dim a load using phase-angle control.
Use the MOC 3041 if you want on-off control at zero-cross to minimise EMI (electro-magnetic interference).