Can a transistor have negative Q point?
As a rule of thumb, you should never have a negative \$V_{CE}\$ in your DC biasing. If you do, the emitter acts as a collector and the collector acts as an emitter. (They're not designed for that, so it won't work very well.)

In your circuit, there's no way to get a negative \$V_{CE}\$. The emitter is connected to the lowest voltage in the circuit (ground).

What actually happens when you increase \$I_B\$ is that the collector voltage drops until it reaches \$V_{CEsat}\$, which is about 0.2 volts. Then it doesn't drop anymore, regardless of how much current you pump into the base. In other words, the collector current is limited by the resistor. You have 10 volts and 220 ohms, so you'll never have a collector current higher than about 45 milliamps.