There is no standard notation called Q (matrix).
However in this case I think the matrix that they are referring to is the state transition matrix (similar to the adjacency matrix as mentioned by Justas in the comments, but with rates instead of just the connections). Basically you have three states (lets call them A, B and C) and there is a rate for transition from one state to the other. You can also represent these rates in the form of transition probabilities.
You just represent this as a 3×3 matrix. Put zero where there are no such transitions (for example there is no transition from A to A or A to C):
& A & B & C \\\hline
A & 0 & 10 & 0 \\\hline
B & 100 & 0 & 50 \\\hline
C & 5 & 0 & 0 \\\hline
You can then calculate the eigenvalue for this matrix. How to calculate eigenvalues and what is their significance is off-topic in this site. You can easily find out how to calculate eigenvalues. Their significance is something that is not that easy to understand but you can read more about that and can perhaps ask a precise question in Mathematics Stack Exchange. Basically, they tell you how the system proceeds in different directions (denoted by eigenvectors).