If you don't have multiple attacks from your BAB but you have a double weapon like a quarterstaff, do you get multiple attacks if you take a full-round action or do you still need the bonus from the BAB for that to be valid?
Yes, a creature that wields a double weapon and that takes the full attack action can make 1 extra attack
Even if a creature lacks the feat Two-weapon Fighting and the creature's base attack bonus is less than +6, when wielding a double weapon and taking the full attack action the creature can make 1 additional attack. That's because according to the Combat Section on Double Weapons...
You can use a double weapon to make an extra attack with the off-hand end of the weapon as if you were fighting with two weapons. The penalties apply as if the off-hand end of the weapon was a light weapons. [sic]
But without the feat Two-weapon Fighting or similar ability the standard penalties for two-weapon fighting apply: a −4 penalty on the attack roll with the first attack with either head of the double weapon, and a −8 penalty on the attack roll for the second attack with the other head of the double weapon.
So while a creature untrained in the art of two-weapon fighting could make that extra attack, it probably shouldn't unless circumstances are really stacked in its favor.
The comment from the anonymous user misrepresented the penalties for two weapon fighting a bit...
Completely untrained and not using a light off-hand weapon, the penalties to attack are -6 (primary) and -10 (off-hand). If your off-hand weapon is light the penalties are reduced to -4 and -8. With the Two-Weapon Fighting feat your primary attack is the same but your off-hand attack improves to also only being at -4. Finally, with the Improved Two-Weapon Fighting feat those penalties decrease to -2 each.
Double weapons count as wielding a light off-hand weapon when using them for TWF, so untrained the penalties are -8 and -4. If you want to go the TWF route with double weapons I would highly recommend the TWF feats. There are some other ones that might compliment your build as well like Double Slice and Two-Weapon Rend.