So my character, due to story reasons, just learned the cantrip Virtue (unearthed arcana). I found a copy of the cantrip here https://www.dndbeyond.com/spells/2005803-virtue, but I do not know how to add it to my character without unlearning a previous cantrip. 

You would have to add the spell to a homebrewed feat and then add the feat directly to the character sheet under Features & Traits->Feats->Manage Feats.

Can you add homebrew spells to a feat? I do not see the option to add the virtue cantrip.

Only if you wrote it yourself.

Instead of adding the cantrip by name, you can just limit by level (cantrip). Then when adding the feat to the character you/the player can simply choose the cantrip, even if the cantrip is public homebrew you added to your collection.