Neuron diversity is great, especially if you include the neurons of invertebrates. As a result there is no simple answer to how many neurotransmitter molecules it takes to trigger a neuron to make an action potential.
The relationship between transmitter amounts affecting a particular neuron and its production of action potentials depends a lot on the geometry and physiological properties of the neuron, as well as where its inputs are located on the neuron.
Also not all neurons even generate action potentials. Some are
non-spiking, just having graded changes in membrane potential.
Among neurons producing action potentials, there is usually a single critical point (the
axon hillock) where the graded changes in membrane potential resulting from synaptic inputs will either exceed a threshold and trigger an action potential or fail a dissipate.
How the various inputs, as discussed above, sum up together is often analyzed by modeling with the neuron's
cable properties. This takes into consideration many properties of the neuron and where its various inputs are. Among the neuron's properties considered are the length and electrical properties (capacitance of membrane, how well different parts are electrically connected), of the number, size and length of neuronal branches (such as dendrites) as well as the cell body, location, density and type of synaptic inputs. This allows the neuron to be modeled in a geometrically simpler form in an attempt to figure out what it takes to generate action potentials in particular neurons.
https://www.amazon.com/dp/019515956X/?tag=pfamazon01-20 is a classic book on this approach that explains this stuff better than I could.
Here is a pdf of the first chapter which explains a several concepts.