Yes you actually can. It's known that HDMI is TMDS and that the fastest frequency on any given pair is 680MHz and there are a total of 13 data pins (4 pairs + i2c + CEC pin + hot plug detect pin + reserved pin for some special features). A digital logic analyser that can sample at that rate over all 13 pins is going to cost less than a grand. If you stub in some hardware to convert the differential pairs back to a single hi/low signal and drop the optional features of the reserved pin, you can cut that down to 8 signals (or less if your analyzer has dedicated clock signal pins). A DSLogic U3Pro16 is 299usd and can sample 8 signals at 1GHz in buffer mode or 3 pins at 1GHz indefinitely in streaming mode.
If you know roughly what you are looking for, you can set triggers to start sampling when the event you care about starts, that's more than enough to be able to reverse engineer even the most intensive of the existing HDMI spec.
Given that a lot of these graphics cards cost substantially more than 300usd, it's not unreasonable to expect a logic analyzer capable of digesting HDMI to be within their price range.
