No, it's nothing to do with Windows and Linux (I assume you mean your DATA, not object names).
What you need to do is set the collation appropriately on your database.
If it's set case-sensitive, it will be case sensitive, otherwise it will not be.
e.g. setting the collation utf8_general_ci will mean that your data are in utf8 and case insensitive.
This is also related to getting your encodings right. Ensure that you do so. Always.
Mark