Nope, here your first degree is your undergraduate degree than all future work is postgraduate. You can be a graduate (obviously) but the term "graduate degree" would make absolutely no sense.
So and Masters or PhD student would always be a postgrad.
Just to confuse matters, it's very rare for any postgraduate degree to ever be a Bachelors so the few that are (such as architecture) as always specified as a post-graduate Bachelors Degree, i.e. I hold a BSc in Architectural Studies and a postgraduate Bachelor of Architecture.
Weirdly enough Oxford University philosophy department seem to use the terms pretty interchangeably:
http://www.philosophy.ox.ac.uk/admissions/gradintro.shtml
Their masters-equivalent degree being a BPhil, just to confuse things.
But there again, Oxford will give all their BA graduates an MA if they pay an extra fee something like a year later.