We can think of the Compiler as a function from a string to a string - high level (HLC), to low level code(LLC). LLC can include the garbage collection code(if it is run as a standalone executable instead of garbage collection being done by a separated runtime).

The compiler executable itself is running in a compilation process P which uses memory and has its own garbage collection. (The compiler executable was itself generated by a compilation, using a compiler written in Go itself(self-hosting) or initially, in another language).

But the compilation process P is unrelated to the process Q in which the generated code, LLC, will run when first executed. The OS which runs LLC doesn't even know about the compiler - LLC is just another binary file. The garbage collection in P doesn't affect garbage collection in Q.

Indeed, it should be easy for the compiler to generate an assembly program which constantly keeps allocating more memory until the system runs out, while compiling say a loop which allocates a struct within a loop running a billion times. Unless, of course, you explicitly also generate a garbage collector as part of the low level code.

Your question does become very interesting in the realm of security, there is a famous paper called "Trusting Trust" where a compiled compiler can still have backdoors even if the compiled code is trustworthy and the compiler code is trustworthy but the code which compiled the compiler had backdoors.