Some of them probably did! But it’s hard to say for sure, so let me try to explain why…
Historians have been interested in the origins of the First Crusade for, well, centuries really - probably as far back as the time of the crusade itself. Did people go to capture land and win glory? Were they simply barbarians looking for somewhere new to expend their violent urges? Perhaps it was a political/military reaction, although a rather delayed one, to the 7th-century Islamic conquest of Jerusalem. Among historians today, probably the most popular theory is that crusaders were mostly pious believers who honestly felt Jerusalem should be part of the Christian world.
Lately, some historians have been focusing more on preaching and prophecies about the apocalypse in connection with the crusades. Unfortunately for us, no medieval author explicitly says “we think the crusade fulfills prophecy in Revelation, and here’s why”. In this case it’s not a very simple historical question, it’s more about historiography - how we should read and interpret the medieval sources, to see if we can figure out what the authors believed and how that fit into their culture and experiences.
For example, the story of the massacre in Jerusalem when the crusaders captured it in 1099 has usually been considered evidence that it was a blood-soaked frenzy of violence. But what if it’s not a literal description of what happened? What if the authors who wrote about it were trying to describe it in terms that would be familiar to them and their medieval readers? Descriptions of rivers of blood and mass executions of prisoners probably come from similar stories in the Bible.
Sometimes the medieval authors also describe “six battles” during the First Crusade, implying that there would later be a seventh and final battle. The number seven often shows up in Christian apocalyptic prophecy, so they must be referring to Biblical stories that everyone knew, without having to spell it out in detail.
One historian of the First Crusade, Fulcher of Chartres, remained in Jerusalem afterwards. Sometimes he wrote about natural phenomena, plagues, locusts, earthquakes, storms and comets and other strange things appearing in the sky. Did he suddenly become interested in nature, or are these Biblical references too? He doesn’t mention specific prophecies from Revelation, but natural events are part of the prophecies in Revelation, and he could be implying that believed similar things were happening.
This kind of reminds me of Redditors referring to memes or quoting TV shows or movies. We know what we’re talking about, but if historians in a few hundred years found a bunch of 21st century memes, would they have any idea what was going on? I think it’s the same idea here. We'll quote Star Wars or the Simpsons; they were using Biblical imagery as a common reference instead.
So, medieval authors don’t really specifically mention Revelation 21, but sometimes we can see that they’re probably referring to it and other parts of Revelation. Some crusaders, but not all, believed they were fulfilling prophecies and their new kingdom really was the Biblical New Jerusalem.
Here are some sources that will hopefully help explain better:
Jay Rubenstein, Armies of Heaven: The First Crusade and the Quest for Apocalypse (Basic Books, 2011)
Jay Rubenstein, Nebuchadnezzar’s Dream: The Crusades, Apocalyptic Prophecy, and the End of History (Oxford University Press, 2019)
Elizabeth Lapina, Warfare and the Miraculous in the Chronicles of the First Crusade (Pennsylvania State University Press, 2015)
Jonathan Riley-Smith, The First Crusade and the Idea of Crusading (University of Pennsylvania Press, 1986)