GNU Radio QPSK 4 Symbols
It is important for the simulation that I'm running that there are only 4 possible modulation values before noise is added to the system.


So, you made a mistake when building your simulation, or your simulation already includes a frequency offset. The way you ask this, you probably did the former, but since we have zero knowledge of what exactly you've built: Can't help you with that, make sure your receiver simulation does exactly the same as what your transmitter simulation does. ("fix your program!")

I'll answer as if this was a system that's generally correctly designed:


  Thus, when using a QPSK modulator in GNU Radio I would expect to see, in the constellation diagram, a square with 4 points at each corner, but instead I see diamonds with thousands of points, even in the absence of any introduced noise. 


That's not surprising! No two oscillators are exactly the same, and so your receiver and transmitter have a frequency offset.

A frequency is the linear change of phase over time. That means that your symbol phase at the receiver has a phase offset that grows constantly every time you look. Hence, the "diamond".

You need what you've been taught in your digital communications course: a frequency synchronization!

You can do that with a second order costas loop.