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This paper investigates the effect of the linewidth enhancement factor on the operation and intensity noise fluctuations of 980-InGaAs/InP fiber grating lasers operating under strong optical feedback. The analyses are performed in terms of the temporal trajectory of the laser intensity, bifurcation diagram and relative intensity noise. The influence of intrinsic fluctuations in the intensity and optical phase of the lasing field on the laser dynamics were taken into account. The laser diode operates in pulsation under strong optical feedback and at small and relatively large values the linewidth enhancement factor. The optical feedback noise is found to be as low as the quantum noise level when the laser is operated in pulsing region and at relatively large values of the linewidth enhancement factor.