There are levels of time definition., and all levels depend on change . Time is a mathematical construct of our mind abstracted from our biological experience.
The first level of time definition comes because we are living observers. We observe changes in our surroundings and ourselves. The changes we parametrize as "time" are strongly coupled in the biological cycle. Living beings have a timeline from birth to death. The giant astronomical clocks were used to correlate these changes with the timeline, defining time: day/night tides sun cycle. As our scientific observations progressed time has also been defined as vibrations of specific atoms, atomic clocks.
This time has as a beginning the emergence of homo sapiens who could organize a concept of dimensions and time. This time cannot be defined in a universe of two particles.
The next level of time is when we learned thermodynamics and statistical mechanics and discovered a microcosm of atoms and molecules and the concept of entropy, which can never become less and it can only grow. This direction gives also a direction for the so called "arrow of time". The number of microstates describing a system can only grow. In this definition of time's arrow two particles cannot create statistical quantities so there is no meaning to time other than mathematical.
Observation of change is important to defining a concept of time. If there are no changes, no time can be defined. But it is also true that if space were not changing, no contours, we would not have a concept of space either. A total three dimensional uniformity would not register.
In special relativity and general relativity time is defined as a fourth coordinate on par with the three space directions, with an extension to imaginary numbers for the mathematical transformations involved. The successful description of nature, particularly by special relativity, confirms the use of time as a coordinate on par with the space coordinates.
It is the arrow of time that distinguishes it in behavior from the other coordinates as far as the theoretical description of nature goes.
Your gedanken experiment of only two particles in a universe can only be a mathematical exercise, the time it has will be the time in the mathematical equations describing your creation and the forces involved. As was observed in the comments static means no time dependence and the problem should be better defined. If you have one hydrogen atom in the universe, that is static, until one goes to the nuclear dimensions, then a mathematical time is need to describe the system.
