There is no direct train nor bus between Obidos to Sintra
I emailed the Tourism Office in Obidos this afternoon and got this reply:
"mas não há nenhum autocarro de Óbidos com
destino a Sintra, sendo que terá sempre que passar por Lisboa"
Translation: "..but there is no bus service from Óbidos with a destination of Sintra, therefore one must always route via Lisbon"
So, incredible as it may seem, no direct public transport service between the two.