Until end of turn effects and Doubling Season
Your friend is not correct. As you identify, Doubling Season's second ability only cares about counters being placed on permanents you control.

If your opponent put a -1/-1 counter on a permanent you control, Doubling Season applies and two -1/-1 counters are placed instead.

Quillspike's ability cost doesn't place counters on permanents. It removes them. Quillspike's ability doesn't place counters, it just gets +3/+3 until end of turn. Neither part of this ability interacts directly with Doubling Season.

However, because Quillspike is "fuelled" by -1/-1 counters, Doubling Season does ensure that you get double the fuel (as long as the creature receiving the counters isn't reduced to 0 toughness). You still have to pay the mana costs to activate Quillspike's ability more than once.