From what I read in my introductory macroeconomics textbook, central banks can control the money supply by selling government bonds in the market (decreases money supply), or by buying government bonds from the market (increases money supply). Countries that have no debt have no need to issue bonds. In the case of these debt-free countries, how do their central banks control the money supply?
There are several ways how central banks can still control money supply even if we assume the government does not issue any bonds.
First, there is the discount rate at which other banks can borrow money form the central bank. The nominal interest rate throughout the whole economy depend on the discount rate as the rate at which banks are willing to borrow to households and businesses also depend on the rate which they can borrow from the central bank.
This is what the Norway (a country with negative net debt) is doing. According to the Norges Bank
Instruments and implementation The main monetary policy instrument is the policy rate, which is the interest rate on banks' deposits in Norges Bank up to a specified quota.
The policy rate is set by Norges Bank's Executive Board. Normally, the Executive Board makes monetary policy decisions eight times a year. The Executive Board meetings where monetary policy decisions are made are called monetary policy meetings.
Second, there is the reserve requirement. In principle central banks could increase money supply by decreasing the reserve requirement which would increase the size of money multiplier $1/RR$ where $RR$ is the reserve ratio. This is quite crude way ho to conduct monetary policy though and its quite uncommon.
Furthermore, as was correctly pointed out in the comments even government with no net debt might still issue bonds, so the assumption that they do not is not necessarily correct. Hence in practice that option is still there at least in principle - although as correctly pointed by @BrianRomanchuck your question excludes it, its worth while to keep this in mind.
Hypothetically, there could also be other ways how this could be done but they are not used too much in practice. For example. there are also precedents for central banks buying up private bonds but its not common. Central banks could in principle also directly change the monetary base with something like helicopter money (depositing money directly to private accounts and hence increasing the monetary base - or in cases like ECB where central bank also has power to directly coordinate printing of the money by literary printing them and throwing them out of helicopter), but since your question is more about how its actually done not how hypothetically it could be done I wont go into them.