This is completely unofficial, basically my own idea of what occurred following the movie. However, for the time being, I will continue to take this as the account of Nick, Judy and their continuing story. Of course, they strike up a romance. I use ideas that I thought up in Loss of Heroes, though I decided that that won't be canon to this. If you want to use any of my ideas, just ask. Enjoy!
About the Zootopia Expanded Universe
Established by fan-writer PrinceBalto on November 4th, 2016, eight months after Zootopia's theatrical release and five months after its DVD release, the Zootopia Expanded Universe, or ZEU, is an unofficial expansion/continuation of Zootopia's story, unlike his 2.0 canon, which is a remake. It was inspired by the famed Star Wars expanded universe (what Disney now calls Star Wars Legends and called a separate, non-canon continuity). Being a shipper of WildeHopps, PrinceBalto established that a key part of the ZEU is a deep, passionate romance between Nick and Judy, starting to take root while they are cracking the case and becoming a full romance just before his ZPD graduation. This continues on with years' worth of adventures, including them getting married. Another expansion is that it gives the names of Nick's parents, Robin and Marian.
The ZEU uses the savage crisis as the starting point for its dating system. Events before the movie are dated as "BSC", or Before the Savage Crisis, or ASC, or After the Savage Crisis.
The ZEU, like the Star Wars EU, goes back many years before the movie to many years afterward. Stories set before involve primarily the childhoods of characters. Stories set after generally focus on the characters afterward, oftentimes Nick and Judy.