The simple answer (less I'm mistaken), no. Generally it's not a good idea for 2 reasons. First is because it causes a trust problem between your program and other programs (not to mention us humans won't trust your application either). second is if you were able to access another applications memory and make a change without the application knowing about it, you will cause the application to crash (also viruses do this).
A garbage collector is called from a runtime. The runtime "owns" the memory space and allows other applications to "live" within that memory space. This is why the garbage collector can exist. You will have to create a runtime that the OS allocates memory to, have the runtime execute the application under it's authority and use the GC under it's authority as well. You will need to allow some instrumentation or API that allows the application developer to "request" memory from your runtime (not the OS) and your runtime have a way to not only response to such a request but also keep track of the memory space it's allocating to that application. You will probably need to have a framework (set of DLL's) that makes these calls available to the application (the developer would use them to form the request inside their application).
You have to be sure that your garbage collector does not remove memory other then the memory that is used by the application being executed, as you may have more then 1 application running within your runtime at the same time.
Hope this helps.