Measuring the I and Q components of each of the two signals only makes sense if there is some third signal acting as the reference. But determining I and Q with respect to some third signal is no easier than determining I and Q of one of the original two signals with respect to the other signal acting as the reference.
If the two signals are very clean (with no noise added), you can transform them from analog sine waves into logic square waves by a threshold comparator where the threshold is somewhere near the zero crossing. Then the XOR method mentioned earlier can be used.
If the two signals are not very clean, you can still extract precise phase information, but then you will need to use something like a Phase-Locked Loop to generate a logic version of the analog signal. A PLL is able to find the average phase of a signal whose phase as determined by simple thresholding might be unstable.
I don't know what building blocks you have available in your FPGA, but if it is only logic building blocks, you will need at least one analog block to perform the comparator threshold function or the PLL function.