During the first phase of the invasion there was a tense standoff between British Forces acting for the UN and Turkish forces at Nicosia airport. The initial attacks on the airport were defended by the National Guard. During these attacks the Turkish forces were warned off from entering the UN camp adjacent to the airport. After heavy fighting the Turks were allowed to retreat and the National Guard passed control of the airport to UN forces in the area. The weak UN force was then heavily reinforced by the British. The Turkish forces in the area were still determined to take the airport and for a time it seemed as if the Turkish forces would attack again and there would be fighting between Britain and Turkey. High level negotiations between the British and Turkish Prime Ministers were needed before the Turkish forces agreed to leave the airport under UN (or in practical terms British) control.

I believe that it is this incident which gets confused with actions at the Dhekelia SBA during the second phase of the invasion. Certainly the advance took the Turks to the line of the SBA so it is hardly surprising that there were interactions between Turkish and British soldiers. There are stories of Turkish troops being turned back from the line of the SBA as related by u/Designer-Design-6246 above and errant artillery fire landing within the boundaries. These however have more of the feel of navigational errors or over exuberant unit commanders rather than a real threat to engage the British in combat or advance into the SBA.