I just set up the Hot Network Questions room (4:17 PM UTC, 9 February 2019) to keep track of the HNQ questions from today onwards. If you want to search whether a specific question was in the HNQ you need to use the room's in-built search tool.
Say, if you want to check whether this question https://travel.stackexchange.com/questions/131853/can-i-travel-from-belgium-to-uk-after-brexit-with-just-an-identity-card was in the HNQ, search for travel.stackexchange.com/questions/131853 as post IDs are permanent although question titles can change. Leave out the https:// part while searching.
Update 1:
It turned out that the RSS feeds for HNQs which SE provides contains only 30 random questions out of the total 100. Fortunately, @rene had written up a Node.js script (running on Glitch) a few months ago, which returns all the 100 HNQs in RSS XML format! The corresponding feed URL is https://lackadaisical-appeal.glitch.me/hnq/+. Thanks to AndrewT and rene who helped me set this up in the HNQs room.
Another important thing is that chat rooms freeze within 14 days if there are no messages by chat users (feed bots i.e. users with negative IDs don't count) within that period. To prevent that, Floern set up Generic Bot which will post something in that room every few days, in order to keep it alive. A hat tip to them!
Update 2:
We realized that chat room feeds are rate limited, and thus not all of the 100 HNQs were being posted in the room. So with Martin's help I set up individual feeds for each site.