If the integrated area has a custom icon set on its Edit tab, that will be used on the forum list. If it has a custom icon and you wish to use a different custom icon for the integration only, this can be achieved with custom CSS rules.
If you want to use a different default icon for your integrations, you should create a clone of /vault/resources/images at a custom directory like /vault/resources/custom-images and change the Wiki Image Directory Style Property to your custom directory. Then you can change any of the stock images as you wish (such as icon/forum/vw-area-icon-36-new.png and 36-old).
Alternatively, if you want all integrations to have the same icons, you can modify the vw_area_forumbit template (or vw_area_subforumbit for sub-forums) so that it uses your desired icons. This might be necessary to get a consistent look if you are not using XenForo's default style.
Note that in XenForo's default style, no icons are used for sub-forum listings, so wiki areas shown in the sub-forum listings have no icons either.
Because your forum-icons differ based on style, it would not be practical to implement and use a separate integration-icon setting if you want consistency with your style. Instead, one of the above style-related methods should be used.