As the comments point out, there's no measuring velocity from the frame of something traveling at $c$. The question becomes non-relativistic if we take a single reference point not traveling at c and measure the rate of change of displacement (our proper length) with respect to time (our proper time) between the quanta, rather than asking the unanswerable question of relative velocity from the point of view of something traveling at $c$.
If we represent our $d\vec s/dt$ vectors graphically, we're just trying to find a configuration of vertices with equal lengths between all vertices. In two dimensions, this is an equilateral triangle. In three dimensions, it's an equilateral tetrahedron (a triangular pyramid).
So, the number of quanta between which we can measure a rate of change of displacement equal to c is four, one for each vertex.
One example, with three massless quanta and one massive quanta, would be three receding at c away from us along the near edges of an equilateral tetrahedron (60 degrees of angular distance between each of the three paths), and one stationary next to us.
We can also do this with four massive quanta. In that case, again, this is a non-relativistic problem. Set it up by starting us at the center of an equilateral tetrahedron. We fire one bullet towards each vertex at velocity $dr/dt = v$. The displacement vector between each bullet is the long edge of a 30-30-120 triangle with a short edge of length v, so the long edge has a length of $\sqrt {3} v$. Solve for v to get $c = ds/dt = \sqrt{3} v \implies v = c/\sqrt{3}$
Now we can give all four quanta any additional velocity $\vec u$ relative to us that we want provided that $|\vec v + \vec u| < c$. In fact, the first case turns out to be just a special case of the four-massive-quanta case at the limit of
$|\vec v_1 + \vec u| \to c$
$|\vec v_2 + \vec u| \to c$
$|\vec v_3 + \vec u| \to c$
$|\vec v_4 + \vec u| \to 0$
The other possibilities: all four massless, two massive and two massless, just one massless, are no good.
Extra notes
We can measure from the point of view of the massive quanta in this set up, since measuring from the point of view of something is the same as setting choosing a frame from which its velocity is zero. Note, however, that for the case of four massive quanta, our quanta will not agree with us about their relative rates of change of displacement unless we are already in their rest frame. Switching between non-comoving frames means you must lorentz transform, which will change the angles and displacements, squashing your equilateral tetrahedron.
Note that of course the speed limit of the rate of change of displacement between two objects as measured from the frame of a third is 2c, since we can point a beam of light in one direction and point a different beam of light in the opposite direction. The number of things that can have that, obviously, is 2. If we put ourselves at the center of a tetrahedron and shoot a laser out through each vertex, $ds/dt$ of the edges is $\sqrt{3}c$.
