Is nucleosynthesis responsible for the expansion of the universe? Is it just a coincidence that the two major expansionary periods occur close to periods of nucleosynthesis?


Big Bang nucleosynthesis occurred very close to the inflationary period.
Supernova and stellar nucleosynthesis created heavier elements as the stars died, and the universe underwent increased acceleration.


Further thoughts:

The metaphor for the expansion of the universe is often described as an inflating balloon, but could it be equally valid to think of expansion as matter falling into its own gravity well, away from infinity minus other matter within its gravity field. The effect being that matter radiates space itself as gravity pulls itself away from infinity. Just as heavier elements create a stronger gravity field, heavier elements would accelerate away from infinity minus other matter within its gravity field, at a faster rate.  

Eventually, on a cosmic scale, atomic decay would release the confined energy and the energy would return to infinity, creating the appearance of a contracting universe that restarts the Big Bang.  If this idea is reality it would create a geodesically complete cosmology.
Big bang nucleosynthesis did not happen "close to the inflationary period".
inflation happened at about $10^{-36}$s. Nucleosynthisis happened from about 10seconds. 

In relative terms, the ratio of the age of the universe now, to 10 seconds is much much smaller than the ratio of 10 seconds to $10^{-36}$s. And, significantly, nucleosynthesis occurred after inflation, so can't be the cause of inflation.

As for dark energy, nobody has any real model for it. The test for any theory is how well it does empirically. As it stands you have advanced a pet theory, without any mathematical model or analysis.