How does it make sense to talk about, for instance, quarks traveling at near lightspeed while confined to a volume as small as a proton? How does it make sense to talk about, for instance, quarks traveling at near lightspeed while confined to  a volume as small as a proton? The distances are so small, and the speed so high, that the notion of motion over time seems meaningless. Using a looked up value for proton size and quark velocities within that confine,  it becomes obvious that smearing out occurs. The notion of time and distance seems to fade into useless attributes. 
Is this connected with the problem of the inclusion of gravity at the subAtomic scale?
 A: First the inclusion of gravity at the subatomic scale is about the QM level, and about gravitons, how they work, yet has there been no experiment to prove their existence.
Nucleons are bound by the strong force and residual strong force, nuclear force. The nucleons are not bound by the gravitational force of each other and if it would be quantized, it will be much smaller then the scale of the strong and residual strong, nuclear force. If you look at the orders of magnitude difference between coupling constants for gravity, it is so small it is not even measurable, not even for bound atoms.
You say that the quarks are bound into a volume as small ass the proton.
Now quarks are point particles. They have no spatial extension, so what would be small space for them?
It is not true to say that quarks are confined into a small volume of space and thus they cannot move near the speed of light in that small volume of space. Since they are point particles, the volume compared to them is infinite, so they can easily move near the speed of light in that volume of space in a proton.
To our knowledge today, the smallest scale is the Planck scale that makes sense to talk about.
Now compare the Plank scale to the size of the proton, being 0.84*10^-15m.
The Planck length is 1,6*10^-35, or 10^-20 times the size of the proton.
And even the Planck length is infinite compared to the point particle.
Now in that size of the proton, being 10^20 times the size of the Planck length, the quarks can easily travel near the speed of light.
Now you are right about that the time it would take for the quark to travel the size of the proton from one side to the other, so 0.84*10^-15m near light speed (300000m/s), that would take very little time in our world.
But what time in our world is small, is not on the Planck scale, the Planck time is 10^-43s. It is much smaller then the time the quark would need to travel the distance from one side of the proton to the other. 
So the quark has a lot of Planck time to travel inside the proton.
