Did the Mongols ever fight other Mongols internally?
yes. This is actually the norm.
Infighting within the Mongols was widespread and the norm. Not only the Mongols, but also all steppe nomads. It is normal for different nomadic tribes and alliances to clash with each other. Every once in a while, a tribe would unify a large group of steppe nomads under their command to form a large nomadic alliance. These vast alliances would invade neighboring empires and pose a great threat to the settled people and their empires. Gokturk Khaganate and the Mongol Empire are two well-known examples of such alliances.
In the days leading up to the rise of Genghis Khan, the Mongols were in constant conflict. Even people like Genghis Khan fought against other Mongols. In fact, much of his early military career was spent fighting Mongol opponents. It was not until 1206, when Genghis Khan unified the Mongol confederation under his own rule, that the infighting ended.
The Mongol Confederacy was unified by Genghis Khan to form the Mongol Empire.
Nor did this unity (without internal conflict) last long. Only a few decades. The death of Möngke Khan, the fourth khan of the Mongol Empire, ended the era in 1259. Internal conflicts between different groups within the Mongol Empire began almost immediately. The empire soon fell apart, and the Mongol khanates attacked each other.
After the death of Möngke Khan (their brothers), the Torreid Civil War broke out over succession between kublai Khan and the Arik Burke brothers. Then there was the war between Berk Khan of the Golden Horde and Hulagu Khan of the Ilkhanate. There was also the war between Kublai Khan and Kaidu. There were also clashes between the Chagatai Khanate and the Ilkhanate.
The four great khanates after the division of the Mongol Empire
With the division of the Mongol Empire during Kublai Khan's reign, these internal conflicts between the Mongol khanates became more common. Further conflicts would occur within these larger khanates, which would eventually split into smaller khanates. Examples include the conflict between the different khanates of the Golden Horde, or the war between Urus Chagatai (Western Chagatai Khanate) and Mughalstan (Eastern Chagatai Khanate).
There was also infighting between different groups within the larger khanates.