I play on 2 computers. But when I complete some quests on the 1st computer, these quests aren't marked as completed if I use the 2nd computer.

Do I have any ways to merge completed quests on all computers where I play? (except of local replacing of addon files)
Can 'QuestieDev' request a server for checking completed and uncompleted quests and update local DB cache by itself?
Any other ideas?

The database of finished quests is saved per character in a table found in <WOW_FOLDER>/WTF/Account/<ACCOUNT_NAME>/<SERVER_NAME>/<CHARACTER_NAME>/SavedVariables/!Questie.lua.

You can copy the whole WTF folder (or sync it with a cloud service) if your systems use the same settings for WoW itself (resolution, etc.) and have the same addons.

If they differ, you will have to copy/sync the files manually for each character or make a Bash/Batch script to do that for you.