Temperature is an scalar function in this case. It remains invariant at every point. The heat flow forms a vector field. The values and components do transform under change of under change of co-ordinate system:
qa = -k∂aT
In new system of co-ordinates (x'):
q'a = -k∂a'T = -k(∂x'a/∂xb)∂bT = -k(∂x'a/∂xb)qb
Now, if these are co-ordinates in your frame of reference. Sitting in your frame you might see some guy moving. You may decide to model how this guy views the heat flow.
To do this first look at how this 'guy'/'thing' is moving in your frame. Obviously he will move with a velocity as a function solely of time or constant (otherwise we would be discussing a whole family of observers). Let the components of his position in your frame be ra in your frame, which is function only of time or a constant. Now assuming he uses 3D Cartesian co-ordinates his frame co-ordinates will be:
x'a = xa - ra
as ra is a function of time, q'a = qa in this case.
However if your observer decides to use some other co-ordinate system (like circular) then naturally the components of heat flow will be different. I hope this answers your question. Please excuse my lack of proper formatting.