Can I use a custom icon for a quest?

If you want to use a custom icon for quest that isn't an item, then you need to use Resource Loader mod and Custom Icon item.

Place your texture in resources/assets/modpack/any_icon_name.png. The file name and folder names have to be lowercase.

Right-click with Custom Icon item in hand and set path to modpack:any_icon_name.png

Select this item as icon from your inventory.