Open Management Studio with configured connections
I'm not sure of a way to do this 'natively' in SSMS, but I've been using SSMS Boost for years and it has that functionality (and a whole lot more). It's a plugin for SSMS and they have a 'free' community version that just needs to be re-downloaded every few months.

Check here and scroll down until you see Download links (updated on 14-th of March 2017):

After installation, there will be a new menu bar option 'SSMSBoost'. Click on the menu bar and select 'Settings'.  Then look for 'Preferred Connections'.  There, you will have a checkbox to open a defined connection on SSMS startup.



You can 'add' preferred connections by clicking the 'plus' sign at the bottom of the screen.  Preferred connections also allow you to specify an 'Important DB Alert' that will actually be displayed on new query windows - I use mine to inform me I'm in a 'production' database.