SelectQuotedText is a plug-in for Notepad++ that selects the text in quotes.
SelectQuotedText is available in the Plugin Manager of Notepad++. That is the easiest way to install it.
To manually install the plug-in, copy SelectQuotedText.dll to the plugins/SelectQuotedText directory. Then (re)start Notepad++.
You can download version 1.1.0 here.
The source code of SelectQuotedText can be found at GitHub.
When the cursor is in a quoted text, press Alt+' to select that text.
You can adjust the hotkey in Settings, Shortcut Mapper, Plugin commands.