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.