Instantons and their quantisation in pure Yang-Mills theory formulated in the background of de Sitter space-time represented by spatially closed (k = +1) Friedmann-Robertson-Walker metric are discussed. First, explicit instanton solutions are found and next, quantities like Pontryagin index and the semi-classical approximation to the intervacua tunnelling amplitude are evaluated. The Atiyah-Patodi-Singer index theorem is checked as well by constructing explicitly the normalizable fermion zero modes in this de Sitter space-time instanton background. Finally, following the kink quantisation scheme originally proposed by Dashen, Hasslacher and Neveu, the quantisation of our instanton is performed. Of particular interest is the estimate of the lowest quantum correction to the intervacua tunnelling amplitude arising from the quantisation of the instanton. It turns out that the intervacua tunnelling amplitude gets enhanced upon quantizing the instanton. PACS 11.10.Cd - Axiomatic approach. PACS 11.15 Gauge field theories. PACS 11.20.Cv - Fundamental problems and general formalism.