The answer is yes, if the Fed purchases an asset directly from the government. But this rarely happens. I could see it happening in perhaps a convoluted bailout scheme with the treasury department.
A good question would be why not, if so many of the Fed's assets are treasury securities. Wouldn't it be smarter to just cut out the middle man and buy directly from the treasury? I think so, but the Fed fears it would threaten its independence and would be less able to control liquidity in the banking system.
Now the Fed does purchase US notes and US coins from the treasury department using "electronic dollars". When the treasury mints a coin that can actually be thought of something perhaps closer to what your question was asking. The treasury is unique in that the money they create abides by different rules and processes. They don't have to buy securities. When the treasury makes 1 billion in coins, it roughly gave itself 1 billion dollars. They then sell this to the Fed for say 1 billion in electronic dollars (or deposits at the Fed). Largely speaking the treasury only prints US notes and coins as demanded by the Fed so this doesn't effect monetary policy, but this hasn't always been the case.