How to square a fixed point number?

Q2.14 is just Q2.14 with implied sign bit. I think the non-implied equivalent would be a 3.14 2's complement number (-4.0 to +3.999).

Multiplying 2 Qm.n fixed point values give an answer of Q(2*m).(2*n). So your output would be Q4.28. The fixed point equivalent of this would be a 5.28 number (-16.0 to +15.999).

In VHDL, I believe you can just infer things directly:

signal left : signed(1 to 17); --Q2.14, 2's comp 3.14

signal right : signed(1 to 17); --Q2.14, 2's comp 3.14

signal ans : signed(1 to 33); --Q4.28, 2's comp 5.28

ans <= left * right;