Circuit Analysis - Function of BJT?
I'm not sure, but it seems they are used to switch off Q1 quickly. Follow my reasoning and see if it makes sense to you.

First of all you should consult the datasheet for the controller chip BQ2031. It describes the chip operations and tells that its MOD pin is the PWM output that allows to control the charging cycle through (ultimately) Q1.

At page 10 you'll see the formula for the frequency of operation, that will depend on the value of C12 (see full schematic in the app-note): 1000pF=1nF sets the frequency at 100kHz, i.e. a period of 10us. This is important because at that frequency C4 can be considered a short circuit: in fact when MOD is LOW and Q4+Q5 are off C4 charges through R4, Q2's base, R6 and R21 (then we reach the output which is at ground for the signal): a total of ~40kOhm. This makes an RC time constant of C4 x 40kOhm = ~ 40us, much longer than the PWM period (the discharge follows a different path, but the resistance seen by C4 is similar).

Therefore we can consider C4 a short for the PWM signal. So we can see that Q2 and Q3 have a complementary function relative to Q4+Q5: these latter turn Q1 on by switching its gate to ground, whereas Q2+Q3 turn Q1 off by switching its gate to "+" (and discharging its gate capacitance quickly). 

The fact that Q2 and Q3 have the same part numbers as Q5 and Q4 (respectively) can be seen as a clue of their complementary action.