If I am a Michigan resident, but sold a home in Tennessee for a gain. Tennessee has no capital gains tax. Do I owe tax on the capital gain in Michigan?

yes as a resident of Michigan 

If you have a capital gain as a Michigan resident, then your return would include a MI-1040D.  If TurboTax picked up the sale from your federal return and transferred it to MI, then you would just need to walk through your MI questions and answers to be sure that if there are any questions regarding it, you answer them.

You can see if MI 1040D is included by going to your print center or Delete a form onlineOn the menu bar on the left that shows. 

1. Select Tax Tools 
2. On the drop-down select Tools 
3. On the Pop-Up menu select Delete a Form 
4. This will give you all of the forms in your return 
5. Scroll down to the form you want to delete 
6. Select the Form 
7. Click on Delete. 

Always use extreme caution when deleting information from your tax return.  There could be unintended consequences. 

Or if you are using the desktop version, just switch to forms mode.