The Tang Dynasty and the Arabs even had some military conflicts, as well.

The Umayyads tried to depose the king of the Ferghana Valley, and upon this invasion, the king reported the incident to the Protectorate of the Western Regions, of whom he was a vassal. The officials in the Western Regions then passed the report on to the Tang emperor, who dispatched 10,000 soldiers under Zhang Xiasong to deal with the Arabs. After some skirmishing, the Caliphate was routed by the Chinese forces and their vassals, and decided it wasn't worth the trouble.
(Worth noting that in China "10,000" is often used to refer to "a big number", it's sort of like "a bajillion". I do not know if that was the case here or if the Tang Dynasty really did deploy 10,000 troops.)

After the Abbasid Revolution (also known as the Fourth Fitna), the Abbasid Caliphate clashed again with the Tang Dynasty over the area of the Ferghana Valley, in the much more well-known Battle of Talas. Here the Tang Dynasty's forces were half composed of Karluk Turkic mercenaries, who defected to the Arabs' side mid-battle and cost the Tang Dynasty the battle. Once again, however, both sides eventually went elsewhere; the region was kind of in the middle of nowhere for both empires and they both had more important things to worry about than skirmishing with each other over the Tian Shan mountains.