Short answer: no, not with romanian soldiers from Romania.
Long answer:
From the beginning of the conflict in 1914 until 1918, about 650.000 romanians were enrolled in the Austro-Hungarian army, most of them in the XII Korp(Sibiu) and VII Korp(Timisoara).
There were many discussions between the leaders of the Romanian National Party(romanian party from Transylvania) whether to support the monarchy or not. In the end, the romanian leaders told the people to enroll in the army and stand by their oath to the emperor. There was hope that Romania will join the war against Russia, so that romanians wouldn't have to fight against each other.
The romanian regiments fought in the war against Russia in Galicia and they faced horrific casualties. For example, the 51 Cluj regiment had 3400 casualties in the first two months of the war from a total of 4000 soldiers. The 63 Bistrita regiment lost in 6 days of fighting 60% of its strength. The 21 Cluj honveds regiment lost just on 24 august 1914 50% of its strength. During the Brusilov offensive the casualties amongst the romanian regiments were even higher.
During the fights in Galicia, romanians from Transylvania, Banat and Bukovina fought against romanians conscripted in the russian army from Bessarabia. This is the only instance of large scale fighting of romanians against each other tho' I may be wrong.
The hunger and cold, along with the treatment the romanian troops received from their officers(mostly hungarians) made many of them to abandon their posts and surrender to the russians in quite large numbers.
In fact, after Romania entered the war in 1916, the number of deserters increased to 40,000 people. There was even a voluntary force made up of romanian POW's from Russia which were released from camps and joined the romanian army.
Now, as I said, after Romania joined the war, the romanian troops were redeployed on the italian front, mainly because it was the state policy that troops shouldn't fight too close to their own homes. Many fell prisoners to the italian army during the Isonzo offensive. Here is a picture with romanian soldiers from Italy upon their return to Romania in 1919.
So, in the end, no, given the fact that Romania entered the war in 1916, and that Austro-Hungary redeployed the romanian regiments from the Eastern front to the Italian front in order to minimise desertions, one could argue that romanians from Austria-Hungary didn't fight romanians from Romania. However, there are stories about romanians from the Empire fighting against their brethren from Bessarabia in Galicia which were conscripted in the Russian army.
I would like to point out that I'm not a historian, so if there's anything wrong with what I wrote, please feel free to correct me.
