I've one domain like www.testme.com and also at the hosting place there's a database (MYSQL). Now i'm using internet on my small home system and prefer to connect that database using MYSQL Query Browser. Can its likely?

Yes, it is possible but you have to enable remote accessibility MySQL Database. Something similar to this:

GRANT ALL ON foo.* TO user@'1.2.3.4' IDENTIFIED BY 'PASSWORD';

Where foo may be the database title, user may be the username you utilize for connecting towards the database and 1.2.3.4 is the current Ip in your own home.

Clearly, your host company should allow remote connections to port 3306 (usually MySQL operates on this port).