So in season one of Game of Thrones, Ser Jorah Mormont tells Dany that "when Aegon Targaryen conquered the six kingdoms, he didn't do it because he had a right. He did it because he could". This, of course, struck me as odd, considering that there are seven kingdoms. Later though, I think, it is conveyed by Oberyn, that Aegon was unable to conquer Dorne, and it remained an independent state which is why they have Princes and Princesses instead of Lords and Ladies. There is a saying amongst House Martell which goes "the iron throne is made up of the thousand swords and none of them are from Dorne." In the HBO YouTube series "Game of Thrones: History and Lore." We are told that eventually two members of House Martell married into royalty, thus uniting the realm, since then House Martell had been considered the crown's closest allies. So does that mean that the Dornish would rally to Dany's side if she crossed the narrow sea?
To answer this properly would require using spoilers from the books. (I can update my answer with spoilers if need be, but I assumed since this is on the TV and movies se that you're looking for a show answer) . - Updated
Avoiding that, it's reasonable to assume they would rally for the Targaryens. Dany's sister in law was Elia of Dorne (sister of Oberyn and Doran Martell), who The Mountain raped and killed during Robert's Rebellion under Tywin Lannister's orders. This is the reason for Prince Oberyn going to King's Landing and is the ultimate reason that he died.
Obviously due to this the Martells have no love for the Baratheons or Lannisters, and so would likely jump at the chance to join with a powerful leader who could defeat these families and help them get revenge for their sister.
Book spoilers below:
The last book details how Doran Martell has been conspiring with Targaryen sympathizers to overthrow the Baratheon/Lannister rule. He has a secret treaty which binds Dorne to the Targaryen cause through the marriages of his daughter Arianne to a boy claiming to be Aegon (the son of Rhaegar Targaryen and the nephew of Dany) and his son Quentyn to Dany.
"I have worked at the downfall of Tywin Lannister since the day they told me of Elia and her children." -Prince Doran Martell to his daughter Arianne Martell.
"I am not blind, nor deaf. I know you all believe me weak, frightened, feeble. Your father knew me better. Oberyn was ever the viper. Deadly, dangerous, unpredictable. No man dared tread on him. I was the grass. Pleasant, complaisant, sweet-smelling, swaying with every breeze. Who fears to walk upon the grass? But it is the grass that hides the viper from his enemies and shelters him until he strikes." Prince Doran Martell to the Sand Snakes