One good source for this would be the Python documentation. A simple text search through the section on Classes reveals the word "call" being used many times in reference to "calling methods", and the word "invoke" being used only once.
In my experience, the same is true: I regularly hear "call" used in reference to methods and functions, while I rarely hear "invoke" for either. However, I assume this is mainly a matter of personal preference and/or context (is the setting informal?, academic?, etc.).
You will also see places in the documentation where the word "invoke" is used in refernce to functions:
void Py_FatalError(const char *message)
Print a fatal error message
and kill the process. No cleanup is performed. This function should
only be invoked when a condition is detected that would make it
dangerous to continue using the Python interpreter; e.g., when the
object administration appears to be corrupted. On Unix, the standard C
library function abort() is called which will attempt to produce a
core file.
And from here:
void Py_DECREF(PyObject *o)
Decrement the reference count for object o. The object must not be NULL; if you aren’t sure that it isn’t NULL,
use Py_XDECREF(). If the reference count reaches zero, the object’s
type’s deallocation function (which must not be NULL) is invoked.
Although both these references are from the Python C API, so that may be significant.
To summerize:
I think it is safe to use either "invoke" or "call" in the context of functions or methods without sounding either like a noob or a showoff.
Note that I speak only of Python, and what I know from my own experience. I cannot speak to the difference between these terms in other languages.