The only way we know that quarks exist is by a series of deep inelastic scatterings with leptons and with protons. This is a reconstructed event at LEP
quark antiquark at 12:00 o'clock and 4:00 o'clock gluon the third one.
The lepton colliders have the advantage that most of the energy taking part in the collision can be detected in four pi detectors.
The transverse momentum, i.e. the momentum perpendicular to the beam is useful, as explained in this answer to a similar question, in the case of hadron colliders , or lepton hadron target experiments because the projections to transverse direction to the beam , due to momentum conservation contain all the energy of the deep inelastic interaction that has gone in the transverse direction. In the beam direction the debris of soft interactions and other possible beam tracks can very easily destroy signals and correlations.
To get the idea about the complexity of hadron hadron events here is a Higgs candidate from the ATLAS detector :
Higgs boson decaying into two τs, each of which subsequently decay into either an electron (blue line) or a muon (red line).
If you go to the link you will see that most of the event displays show the transverse projection, because it is the cleaner from noise, from soft interactions of the three quark to three quark collisions and from extraneous beam particles colliding softly.
So the quark transverse momentum in LHC data will be the momentum of the quark jet perpendicular to the beam. Incidentally, the first indication of the quark content distribution in the protons was given by the existence of high p_t events in deep inelastic scattering of leptons on targets, and this discovery eventually lead to the establishment of QCD as the strong interaction force .