I see the Azure connector that will allow me to pull data from Azure into Domo, but is there a way to do the opposite and push data from Domo into Azure? For example, if I use Domo to pull down data from Google Analytics, is there a way to then push that Google Analytics data from Domo into Azure?
Where there is a will, there is always a way.
Yes, there are a raft of writeback connectors. They are a premium feature, so have a conversation with your CSM / AE about Integration Cloud, https://www.domo.com/integration-cloud.
If you have the technical knowhow you can write your own writeback connector. https://developer.domo.com/docs/custom-connectors/connector-dev-studio
You can also use the JavaCLI to extract data from your instance of Domo and then push it into the cloud. If the datasets are sufficiently large it may make more sense to build a Python script. https://knowledge.domo.com/Administer/Other_Administrative_Tools/Command_Line_Interface_(CLI)_Tool