At 13 weeks your uterus is barely out from behind your pubic bone so no your stomach wouldn't be hard. I'm 35 weeks & baby is very big & my belly isn't hard but I have hard places where baby is laying at that time.
im 13 weeks today, and allready showing. my belly isnt "hard" but i feel flutters of movement usally every day now. but this is my 5th pg.
I'm 13 weeks as well and I hardly can feel anything. I also have barely started to show. Good luck!
The stomach doesn't "harden" per se, what happens is that as the baby grows, you can feel the baby more and more, and he feels firmer than a stomach. So not at 13 weeks, you would need the baby to weigh 5 lbs. or more before mostly what you feel in there is baby and not your abdomen.