The effective photon-quark-antiquark ($gamma q overline{q}$) vertex function is evaluated at finite temperature in the presence of an arbitrary external magnetic field using the two-flavor gauged Nambu--Jona-Lasinio (NJL) model in the mean field approximation. The lowest order diagram contributing to the magnetic form factor and the anomalous magnetic moment (AMM) of the quarks is calculated at finite temperature and external magnetic field using the imaginary time formalism of finite temperature field theory and the Schwinger proper time formalism. The Schwinger propagator including all the Landau levels with non-zero AMM of the dressed quarks is considered while calculating the loop diagram. Using sharp as well as smooth three momentum cutoff, we regularize the UV divergences arising from the vertex function and the parameters of our model are chosen to reproduce the well known phenomenological quantities at zero temperature and zero magnetic field, such as pion-decay constant ($f_pi$), vacuum quark condensate, vacuum pion mass ($m_pi$) as well as the magnetic moments of proton and neutron. We then study the temperature and magnetic field dependence of the AMM and constituent mass of the quark. We found that, the AMM as well as the constituent quark mass are large at the chiral symmetry broken phase in the low temperature region. Around the pseudo-chiral phase transition they decrease rapidly and at high temperatures both of them approach vanishingly small values in the symmetry restored phase.
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