Alright so I'm pretty sure doubling season and like cards don't affect actions related to +X/X until end of turn or -x/X until end of turn because it doesn't physically say counters. My friend on the other thinks the opposite. For instance I have doubling season up, opponent gives me a -1/1 counter it would give my creature -2/2 due to doubling seasons wording, I give the negative counters to Quillspike to get +3/3. Here's where the difference comes in. He believes that for each negative counter given to Quillspike, Quillspike would receive +6/6 until end of turn even though they aren't worded as counters. Is this correct?
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.