Can the passage of time be measured in the absence of motion? All of the ways I have heard of to measure the passage of time involve measuring some sort of motion (e.g. vibrations of a cesium atom, movement of the hands on a clock, etc.).
Can the passage of time be measured without reference to any motion?
 A: Motion is relative. So even if you think you aren't moving, someone else will think you are moving.
So you just have to accept that something measuring time might be moving. And you have to figure out how to correctly handle that.
And it turns out the path of the moving object in 4d spacetime affects how it measures the metric along the path. And that (the metric along the path) is what a clock measures. Not time.
A: To "measure" the passing of time requires that something change (a second hand moves, a crystal vibrates, a counter increments, water falls, light travels, etc). If nothing is changing, then there is no way for you to tell the difference between $T_1$ and $T_2$. Even if you're watching paint dry, superficially motionless, you'll still be mentally counting seconds, and your brain will change as you count chimpanzees.
Now, is what I mean by "change" the same as what you mean by "motion"? Can something change with no motion whatsoever? Although a philosopher might come up with a different answer, as an engineer I would say that all change involves motion. If you look at radioactivity, evaporation, chemical decay, or even gravity waves, all fundamentally involve something moving.  
A: Short answer is no.
Time in itself is not an entity. It is a measure of events in terms of other type of events. The events involve motion at some level (visible, or not). For example, if you have a blinking clock, (instead of oscillating, or rotating), there is no visible motion but there is some kind of underlying motion that regulates the sequence of blinks. The interval itself is in terms of underlying events. Thus, time is a measurement, and it must involve some events, which must have motion/action at core.
Therefore, measurement of time (it's flow) is not possible without involving events (motion).
That also implies that there can not be motion without measurement (flow) of time. Because, detection of motion, implicitly involves other events which take form of flow of time.
We can say, that time flow, and motion individually do not make sense. They both exist together, or neither does. Needless to say, an observer is involved.
The direction of time flow is same as the sequence of events used to measure its flow. Which happens to be always forward, per perception of any observer.
