Problem with utf8mb4 Format when importing database

I wonder if what is happening here is the default database settings are for utf8mb4 if the database server supports it.

In settings.php or settings.local.php if that's where you put your database settings you may have settings that look like:

$database = array(
'database' => 'database_name',
'username' => 'user',
'password' => 'pass',
'host' => 'host_name',
);

if you add in charset and collation then it may stop nagging you to convert. So your new settings would look something like:

$database = array(
'database' => 'database_name',
'username' => 'user',
'password' => 'pass',
'host' => 'host_name',
'charset' => 'utf8',
'collation' => 'utf8_general_ci',
);

Those are fallback settings it utf8mb4 isn't supported.

To go back to your questions:

• if there are errors with Backup and Migrate, raise an issue there: https://github.com/backdrop-contrib/backup_migrate/issues 
• if you encounter errors with the conversion tool, you could raise a core issue: https://github.com/backdrop/backdrop-issues/issues 

With both of those, include as much detail as possible including Backdrop versions, database server versions and other environment information.

There is also a recently added conversion tool within Bee - see https://github.com/backdrop-contrib/bee/wiki/Usage#utf8mb4-convert - though this is based on the core methods so you may not notice any difference.