I import an .ics file. It shows up in my em Client machine fine, but it doesn’t show up on the google calendar until I quit (and restart) emClient even though I hit refresh.
The first thing to check is did you import the items to a Google Calendar. Open Google Calendar in your web browser and confirm if the items are there.
Otherwise, there is a history of eM Client not syncing with Google Calendar. Affects only some users. 
Go to the Calendar section of eM Client, then untick and re-tick the folder your imported the messages to. See if there is any difference.