The production of the stable isotope Li in standard Big Bang nucleosynthesis has recently attracted much interest. Recent observations in metal-poor stars suggest that a cosmological Li plateau may exist. If true, this plateau would come in addition to the well-known Spite plateau of Li abundances and would point to a predominantly primordial origin of Li, contrary to the results of standard Big Bang nucleosynthesis calculations. Therefore, the nuclear physics underlying Big Bang Li production must be revisited. The main production channel for Li in the Big Bang is the H(α, γ)Li reaction. The present work reports on neutron-induced effects in a high-purity germanium detector that were encountered in a new study of this reaction. In the experiment, an α-beam from the underground accelerator LUNA in Gran Sasso, Italy, and a windowless deuterium gas target are used. A low neutron flux is induced by energetic deuterons from elastic scattering and, subsequently, the H(d, n)He reaction. Due to the ultra-low laboratory neutron background at LUNA, the effect of this weak flux of 2-3MeV neutrons on well-shielded high-purity germanium detectors has been studied in detail. Data have been taken at 280 and 400 keV α-beam energy and for comparison also using an americium-beryllium neutron source.