Yes and no. No, it won’t ever happen to our moon, which is actually drifting away from the Earth. BUT it does happen; a really good example of this is Mars’ smaller moon, Phobos. It’s really close (~6000km) from Mars and it’s most likely a captured asteroid made of rubble. We’ve detected long grooves in the surface that are evidence of Martian gravity pulling it apart on a timescale of about 40 million years.
On the other hand if you were really in a sci-fi movie the answer is almost definitely no; if our moon suddenly gets sent crashing towards the earth due to aliens/outside impact/human interference, it’s probably moving too fast to get pulled apart by tidal forces. Some small, loose bits might get pulled into stable orbits but the whole would probably crash first