does the government fund research?

NASA funds little quantum physics research since NASA's mission really is not very closely related to quantum physics. However, there is a lot of funding for quantum research that comes from the government. As others have said, the National Science Foundation (NSF) funds a lot of basic research including quantum research. And the Department of Energy has some funding. So does the Department of Commerce, especially through NIST. And there is a lot of funding that comes from the Department of Defense, as well the intelligence community. There are lots of government funding agencies that fund all sorts of research, some of it being quantum, and these groups have more acronyms than I can name (DARPA, IARPA, and so on). Funding for quantum research comes from a whole host of places within the government. Each department of the government has its own money for funding what they think is important to their missions. And given that quantum is such a broad field, there are many groups that fund it. I would guess that NASA even has some amount that gets spent on it (though not much).

Funding for research in quantum physics or mechanics comes primarily through the Department of Energy, with some additional funding through the National Science Foundation.

NASA and the National Science Foundation are the big funding agencies in the US for astronomy and astrophysics research. For NASA, you often don't just apply to NASA, but to a specific program or satellite fund - so quantum mechanics often wouldn't apply, but, say, gamma ray astronomy might. There are other sources of funding people working in quantum mechanics can apply for, but most will come through the gov't.

The government funds lots of research, even physics research. Public institutions such as universities get government grants and other government funding. Large projects like particle accelerators (such as CERN) get government funding, even international collective funding. Unfortunately, our current president cut funding for a planned particle accelerator early in his administration.