Step 1: Open Settings.
Step 2: Select Accounts.
Step 3: Tap on the Google account you primarily use.
Step 4: Select Account sync.
Step 5: Make sure the toggle adjacent to Contacts is ON.
Step 6: Back up contacts manually by following these instructions:

1 Tap on the 3-dot icon at the top right.
2 Tap on Sync now.

Your contacts and the other information that was selected in the sync page will get backed up to your Google account.