This is a question I've thought and is, to me, could be an interested setting or if there isn't one there can be a dwarf character and a dragonborn character could create this country idea. Thought that clearly depends on if there's a dwarf/dragonborn country already in existence already or not. 

Most D&D nations are either single species racists or multi-cultural with many races.  Hard to come up with a good reason for a country to only have two races.

There are a few with two main races, but it is usually something like 45% dwarf, 33% dragonborn, 10% human, 12% all others.

Quote from Titan_Shadow_Prismsmith >>

This is a question I've thought and is, to me, could be an interested setting or if there isn't one there can be a dwarf character and a dragonborn character could create this country idea. Thought that clearly depends on if there's a dwarf/dragonborn country already in existence already or not. 

 
If you’re the DM, there is if you want there to be.

Depends on the setting  Mystara has the Kingdom of Rockhome,  Krynn Had Multiple kingdoms  Kal Thax, Thorin, Thorbardin, Thoradin, Kayolin and Zhakar.  Forgotten realms has one  Underhome, Greyhawk had Crystal mist.  As far as Dragonborn go i honestly would say probably not 

there is if you want it to be, but really dig into the society. cities with multiple races just let anyone in. cities with only one race are only built and hospitable to one race. you see the problem. if they let two races in, why not the rest? dwarfs and dragonborn don't really have overlapping characteristics, so the city has to be geared toward BOTH of them, and many other races can live in a place geared towards both. do they turn away immigrants of other races? how much of a race do you need to have to be considered eligible? do they have genetic testing? ask yourself these questions