The easiest quickest way is probably to use phpMyAdmin or similar tool. I’ve done it many times and take about 15 minutes which mostly tied up in logging in and out of the two sites and tools.
On the temp server export the Connections tables as a SQL file, just use the default export settings. The tables are easy to identify because they all have “connections” in their name. Then on the new server import the SQL file. If you’ve already installed Connections on the new server, make sure to drop the tables before importing the SQL files.
Hope that helps.