Latin word tertio comes from Latin tertius ((ordinal) third, the ordinal number after and before.), Latin -one(-onem|m)

Detailed word origin of tertio

Dictionary entry	Language	Definition
tertius	Latin (lat)	(ordinal) third, the ordinal number after and before.
-one(-onem|m)	Latin (lat)
tertio	Latin (lat)	(transitive) I do for the third time, thrice. For the third time. In the third place, thirdly. Three times, thrice.