Shortcut to compile the .tex and .bib in one go with Texmaker?

Just use the "User" quick build command (or one of the user commands in the "User" menu).

1. Options -> configure Texmaker -> Quick build : select "User"
2. Click on the "wizard" button and add "pdflatex, bibtex, pdflatex, pdflatex, pdf viewer"
3. Just use the "quick build" command on your document (F1)

As of version TexMaker 4.4.1 (or earlier) the suggestion from user27168 exists as option 2 for quick build