I am attempting to analyze the indexes of several of my tables according to a Microsoft article that describes Reorganizing and Rebuilding Indexes. The article says to right-click on the index within SSMS and then select properties. However, I don't see any option for properties on the right-click menu. I am using SSMS v17.1 with an Azure SQL Server v12.0.2000.8 (which I believe is SQL Server 2014).
This functionality is not available for Azure yet.
You have to use
INDEXPROPERTY. Explained here with example for Azure.
On a side note Azure SQL Server v12.0.2000.8 is not SQL Server 2014. You cannot compare versions between Azure and On-premises. Rather you should compare features if you have to.