It seems like a few related but different concepts were mixed in the movie as well as in your question.
1) Alpha/Be sources
Yes, it is possible to make a neutron source from plutonium and beryllium. Usually, plutonium oxide and beryllium powder are milled and mixed and sealed in a leak-tight stainless steel capsule. The involved reaction is:
$$\ce{^9Be(\alpha,n)^12C}$$
So actually, plutonium is not needed for this; any alpha source with sufficient energy would do. Typical laboratory sources (e.g. for the online boric acid measurement used for a pressurized water reactor) contain $\ce{^241Am/Be}$. The urchin in the first nuclear weapons was a $\ce{^210Po/Be}$ source. $\ce{^226Ra/Be}$ sources are easy to obtain but need more shielding of the gamma radiation.
Since most alpha sources would work, this method would not be suitable to identify weapons-grade plutonium as shown in the movie. Furthermore, because of the short range of alpha radiation, the materials need to be carefully mixed; placing a beryllium rod next to a plutonium sample would not work so well.
2) Primary and secondary neutron sources in a reactor
Yes, nuclear reactors have additional neutron sources. The primary neutron sources are usually $\ce{^252Cf}$ (the most expensive element I have ever handled in visible amounts). They have a half-life of only 2.6 a, but they are only needed for the first cycle. $\ce{^252Cf}$ is a spontaneous fission source. So this topic is not related to plutonium and beryllium as shown in the movie.
Secondary neutron sources are made from antimony and beryllium. During the operation of the reactor, antimony is activated by $\ce{^123Sb(n,\gamma)^124Sb}$. The high-energy gamma radiation of $\ce{^124Sb}$ is used for a different reaction with beryllium:
$$\ce{^9Be(\gamma,n){}2\alpha}$$
Since the antimony needs to be activated first, the secondary neutron sources are not available during the first startup of the reactor.
Although secondary neutron sources contain beryllium, this has nothing to do with the movie.
Note that there is a widespread misconception about such neutron sources. The primary and secondary neutron sources are actually not needed to make the reactor critical. The critical configuration depends on the fuel, moderator, absorbers, and reflectors; but it is independent of the presence of any additional neutron source. And there are always enough neutrons to start a chain reaction. However, the initial neutron count rates are very small; that's why additional neutron sources are needed so that the instrumentation can measure the neutron count rate during the startup process of a reactor with a fresh core. Otherwise, the instrumentation would be blind during first stage of the startup process.
Also note that no additional neutron sources are needed when the reactor contains a sufficient amount of irradiated fuel (i.e. after a partial reload) since irradiated fuel already contains various neutron sources (e.g. $\ce{^242Cm}$ and $\ce{^244Cm}$).
3) Neutron coincidence counting
Yes, it is possible to measure the amount and quality of plutonium (e.g. in weapons-grade plutonium or reactor fuel). In particular, this is done when you get a visit from an IAEA safeguards team. However, this is usually not a single measurement.
Coincident neutron detectors can be used to measure the amount of plutonium in non-irradiated fissile material (like MOX fuel). Neutron coincidence counting is used to
distinguish the neutrons emitted from a single fission event from neutrons created
from other processes. It seems like the measurement shown in the movie was inspired by a passive coincidence detector system.
However, spontaneous fission in plutonium is mainly caused by $\ce{^240Pu}$ and other even-numbered nuclides. Whereas for weapons-grade plutonium, you would be mainly interested in the fissile nuclide $\ce{^239Pu}$. Therefore, the isotopic
abundance must be known, typically measured by means of a high-resolution
γ spectroscopy. Using this isotopic abundance, the total plutonium amount in the sample can be determined from the coincident neutron count rates of $\ce{^240Pu}$.
