What determines the influx of calcium ions in the voltage-gated ion channels?
So, let's use a neuron to neuron connection as an example. 

Calcium is released into the neuron when voltage gated ion channels, specific to calcium, open. These channels can only open when there is a significant depolarisation in the axon terminal of the neuron. 

Where do the calcium ions come from? The majority of them are outside of the cell by the plasma membrane and adjacent to the synaptic vesicles.

I'm not too sure what you mean when you ask what the calcium ion's "source" is. It is common for calcium ions to sit outside the cell, not just in neurons, to help maintain potential difference for example. 

And yes, the influx of calcium in a neuron is determined by the opening of voltage gated calcium channels.

Sources: https://web.williams.edu/imput/synapse/pages/IIA1.htm , https://en.wikipedia.org/wiki/Calcium,  https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3249630/