Going with the Quarter Cask for your first venture into an Islay malt is definitely going all in. Not all peated whiskies will have that medicinal taste, but many Islays will as this is a typical feature of Islay whiskies; especially younger Islays. The Quarter Cask is not aged for a long period of time because it draws flavors much more quickly from the smaller cask (the cask it is aged in is literally one fourth the size of a normal cask; hence the name). Its good to know you're not giving up and trying it different ways in order to enjoy the whisky (even if you added ice).
Personally, the first time I tried the Quarter Cask, I did not enjoy it too much. I had it again last week and to my surprise, I enjoyed it very much.