You can install Groove in a Sandbox through the Appexchange without any limitations. When testing Groove out with your Sandbox, you will need to connect your Groove account and your Groove Chrome extension to to your Sandbox.
Connecting your Groove extension to the Sandbox
On the Omnibar in Gmail, click Log Into Sandbox.
If you're already logged in to Salesforce through your Omnibar, you'll need to log out first, then log back in to your Sandbox.
Click on your Groove extension, and click on the gear icon to find your extension options:
Scroll down to the bottom of the page you're directed to, and click Logout
Right click on your Groove extension in your browser, and select the option Manage Extensions
Toggle your Groove extension off, then back on (to refresh your settings)
Now that you've logged out, go back to Gmail and log into your Sandbox via the Omnibar, as shown above.
Connecting your Groove account to your Sandbox
When using Groove Flow or advanced features you have to login to Groove and connect your Groove account to Google and to Salesforce. In order to connect your Groove account with a Sandbox, click on the link "Connect to a Sandbox" in the Connect Salesforce screen.
If you're already connected to Salesforce, you'll need to disconnect and reconnect to your Sandbox within your Account Settings: