Not at all, because the time-dilation is a local thing, it happens in small regions that don't care about the global expansion of the universe, and it has to do with the speed with which you traverse a closed loop in space, just like the length of a spiral along one turn is different if the spiral is stretched or smooshed. If you have a muon going in a circle fast in a magnetic field, the time-dilation is only a function of the speed, as can be seen by how fast the muon decays. It has no relation to the expansion of the universe, which is only visible on galactic distance scales.
The effect of time-dilation is very simple, it's geometry, except with a different sign in the pythagorian theorem between time and space. You can learn about it here: What are the mechanics by which Time Dilation and Length Contraction occur?