jQuery - find child with a specific class

I'm not sure if I understand your question properly, but it shouldn't matter if this div is a child of some other div. You can simply get text from all divs with class bgHeaderH2 by using following code:

$(".bgHeaderH2").text();

Based on your comment, moddify this:

$( '.bgHeaderH2' ).html (); // will return whatever is inside the DIV

to:

$( '.bgHeaderH2', $( this ) ).html (); // will return whatever is inside the DIV

More about selectors: https://api.jquery.com/category/selectors/

$(this).find(".bgHeaderH2").html();

or

$(this).find(".bgHeaderH2").text();