jmap is the standard java utility that you can use to capture heap dumps and statistics. I can't say what protocol is used by jmap to connect to the JVM to get this info, and it's not clear if this information is available to a program running in the JVM directly (though I'm sure the program can query it's JVM through some socket to get this information).
JVM TI is a tool interface used by C code, and it has pretty much full access to the goings on of the JVM, but it is C code and not directly available by the JVM. You could probably write a C lib and then interface with it, but there's nothing out of the box.
There are several JMX MBeans, but I don't think any of them provide an actual object count. You can get memory statistics from these though (these are what JConsole uses). Check out the java.lang.management classes.
If you want some fast (easy to implement, not necessarily a quick result as a jmap takes some time), I'd fork off a run of jmap, and simply read the resulting file.
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