The Si(1 1 1)5 x 2-Au surface comprises a quasi-one-dimensional surface reconstruction consisting of Au stripes mediated by honeycomb Si chains. In the present study, using STM observations we explored adsorption of thallium (Tl) onto Si(1 1 1)5 x 2-Au surface as well as growth of C-60 layers on Tl-modified and intact Si(1 1 1) 5 x 2-Au surfaces. Analysis of the results was based on the recently proposed KK model of Si(1 1 1)5 x 2-Au surface [Kwon and Kang, PRL 113 (2014) 086101]. It was found that intact Si(1 1 1)5 x 2-Au surface produces a poor template effect on the overgrown C-60 layers which have a close-packed-like structure without distinct ordering. In contrast, the C-60 layers grown on the Tl-adsorbed Si(1 1 1)5 x 2-Au surface where Tl atoms occupy various adsorption sites on the honeycomb Si chains have a well-ordered quasi-one-dimensional structure consisting of straight C-60 chains. Ordering of the C-60 molecules is a result of an intricate interplay between adsorbed C-60 molecules with each other and with a substrate surface. Separation between neighboring C-60 chains is dictated by the periodicity of the underlying Si(1 1 1)5 x 2-Au template, while the C60 C60 spacing within the chains of similar to 12 angstrom that is similar to 3.2a(0) reflects the peculiarity of the C-60-C-60 intermolecular interaction on the Tl-adsorbed Si(1 1 1)5 x 2-Au surface. Structural information on (Au, Tl)/Si(1 1 1) and C-60/(Au, Tl)/Si(1 1 1) systems gained in the present study can be useful for the prospective works on elucidating the ways to control electronic properties of the Si(1 1 1)5 x 2-Au surface using atomic and molecular adsorbates.