I have an Ivy Lane Denizen, how does targeting work?
Cards in Magic: The Gathering (at least when looking at the official Oracle wording, found on Gatherer) usually mean exactly what they say, nothing more and nothing less. If you had to place the counter on the denizen, it would've said "Whenever another green creature enters the battlefield under your control, put a +1/+1 counter on Ivy League Denizen.", and if you had to put the counter on the creature that just came out, it would've said "Whenever another green creature enters the battlefield under your control, put a +1/+1 counter on that creature."

When an ability or spell you play uses the word "target", you are entirely free to choose a target to your liking, within any explicitly mentioned restrictions (it has to be a creature since it says "target creature", said creature can't have shroud or protection from green, etc.)