I know there are build triggers. But do deployment triggers exist in team city? I googled quite a bit and looked at the doc and cannot seem to find it. I basically want custom code to check few things before deploying. But it needs to happen from team city's deploy page.
You should be able to do this by adding a Build, Snapshot, or Artifact dependency. I'm guessing you want a snapshot dependency;
You can then put your tests into that target (or another one in the chain) depending upon your needs. There's really too much to put into a brief answer here. It's best if you read the complete documentation.