Mathematically it implies that the unit vector, i.e. the vector of all 1's, lies in the null space of this Q matrix; this implies some sort of conservation, or balance, e.g charge/mass balance during transition from one compartment to another in a closed multi compartment system. Look for such balance/conservation laws satisfied by the Q matrix in your problem.

I'm not a biologist, but mathematically the matrix makes me feel as follows:

A can change to C, G, or T not to foreign X. [Closed system]

The probabilities of A changing to C, A changing to G, and A changing to T are the same. [Randomness]

When 3a A's are lost in a given time, 1a C's, 1a G's, and 1a T's are generated simultaneously. [Conservation of total number (A+C+G+T) in this term: -3a+a+a+a=0.]

The above is also true for interchanged A, C, G, and T. [Symmetry]