No. Employees of the Federal Reserve Banks are not government employees. They are paid as part of the expenses of their employing Reserve Bank.

Are the Federal Reserve Banks private companies?

The Federal Reserve Banks, created by an act of Congress in 1913, are operated in the public interest rather than for profit or to benefit any private group.
Commercial banks that are members of the Federal Reserve System hold stock in the Reserve Bank in their region, but they do not exercise control over the Reserve Bank or the Federal Reserve System. Holding stock in a regional Reserve Bank does not carry with it the kind of control and financial interest that holding publicly traded stock affords, and the stock may not be sold or traded. Member banks do, however, receive a fixed 6 percent dividend annually on their stock and elect six of the nine members of the Reserve Bank's board of directors.
Although they are set up like private corporations and member banks hold their stock, the Federal Reserve Banks owe their existence to an act of Congress and have a mandate to serve the public. Therefore, they are not really "private" companies, but rather are "owned" by the citizens of the United States.