That's not how the U.S. financial system works. The government doesn't "get money from the Fed" - it gets money from the Treasury, which either collects it from taxes or issues government debt to raise funds.
The Fed participates in open market operations, and one of the markets they participate in is the Treasury bond auction. But they participate in others as well: notably, they've recently been major participants in mortgage-backed securities and corporate debt markets. When they participate, they effectively inject cash into the system, since the amount they can pay is unlimited and subject only to their mandate to get full employment and limited inflation.
There's a lot of other subtleties involved as well; this isn't intended to be an exhaustive explanation.
> When they participate, they effectively inject cash into the system, since the amount they can pay is unlimited and subject only to their mandate to get full employment and limited inflation
In the case where they buy Treasury bonds, the government itself gets money that they can use as they see fit (I.e. stimulus checks), right? Whereas in this case they just "inject cash into the system" in general.
But if the Fed hadn't bought the Treasury bonds, some other investor would've, just at a worse price. That's the thing about markets: participants are fungible, as is the money itself.
There's a big difference in dynamics between one firm giving money to another in a non-competitive situation (say, a union bargaining with a monopsony corporation) vs. a market of buyers trading with a market of sellers (say, the commodity or stock markets). You get efficient price discovery in the latter situation, you don't in the former. And the concern up-thread about the sheer scale of Fed purchases is that when the Fed becomes the only market participant, it starts to look an awful lot like the former situation, and market mechanisms break down.
The government gets money from the Fed by selling them bonds in exchange for cash.