Not at all - it is the "ADR DISAGREE" status that triggers Alternate Law 2, not the stall or overspeed warnings. Based on what I've been told, one should not hear the stall warning in Normal Law before the protections limit the aircraft AoA, but it is possible to hear the overspeed warning before the AoA protections take effect.
Somewhere back in the very early AF447 threads it was said that it was possible to stall an A320 in Normal Law, but it required a very specific combination of (IIRC) gear down and asymmetric flaps - possibly a couple of other things.
The short answer is in Normal Law the AoA should never extend beyond the safe parameters in all but a single specific (and extremely unlikely) circumstances. This is different from the older A300/A310 generation, where the limits were more basic and it was possible to end up "fighting" the aircraft, which may be what the OP was thinking of.
A UAS situation that lasts for more than (IIRC) 30 seconds engages and latches Alternate Law 2, where all protections can be overridden by pilot input.