Basic Transistor Transconductance Amplifier - Negative Current Gain?
The issue is the direction of current. Let's consider a short-circuit load connected to small-signal ground, which is often used to characterize system transconductance:



A small-signal increase in base voltage will induce a larger conventional current flowing into the collector, through the BJT, and to ground out the emitter. As a result, a negative small-signal current is applied to our short-circuit load (using the sign convention shown in the image, which points \$i_{out}\$ toward small-signal ground, from the output. The discussion in your text happens to use this particular convention.

On another note, be careful with your example. \$\Delta V\ = 0.5\,[\text{V}]\$ is not a small signal and the linear model is a poor model of behavior under such an excitation.