I was finally able to figure this out by following the directions at Re: Can you change Estimate to show Quote in quick... (intuit.com) 

First, I had to go into an "Estimate" and select "Old Layout" in the top right corner.

Then I followed the directions in the link above.

(I changed the template name to Quote so I could easily identify it in the "Customize" step below)

Once that was complete, I saw that "Estimate" was no longer an option in the Sales drop down menu.

To find the updated Quote form I selected "All Sales" in the Sales drop down menu

Then I found "Estimate" in the New Transactions drop down menu

At the bottom of that form, I selected the "Customize" link and chose the Quote template from the list

Then I selected "Print or Preview" and was able to see the new template was applied

I was also able to see the Quote template when I selected Save and Send.

Hope this helps others!