The environment of a quantum dot that is connected to two leads is modeled by telegraph noise, i.e., random Markovian jumps of the (spinless) electron energy on the dot between two levels. The temporal evolutions of the charge on the dot and of the currents in the leads are studied using a recently developed single-particle basis approach, which is particularly convenient for the averaging over the histories of the noise. In the steady-state limit, we recover the Landauer formula. At a very fast jump rate between the two levels, the noise does not affect the transport. As the jump rate decreases, the effective average transmission crosses over from the transmission through a single (average) level to an incoherent sum of the transmissions through the two levels. The transient temporal evolution towards the steady state is dominated by the displacement current at short times, and by the Landauer current at long times. It contains oscillating terms that decay to zero faster than for the case without noise. When the average chemical potential on the leads equals the dot's "original" energy, without the noise, the oscillations disappear completely and the transient evolution becomes independent of the noise.