Good question. The answer is yes depending on your aims.
To a Server
Everything in HTML and Javascript is client side, that means, whoever is playing your game can theoretically view all the source to it. It will be obfuscated as part of the export process which will serve well in certain types of plagiarism, but for storing connections to databases, this is a big no no as anyone that can view the source and extract the database username and password could cause havock on it by executing any command they wanted. This is why there is no native support for database connections in HTML/JS.
To Local
However, HTML5 has introduced Web Storage, which allows you to create a database on the clients machine. More information is available at:
http://openbit.co.uk/?p=135
This will eventually be supported by Construct, but it will probably only be suitable for save games etc. No server interaction.
Resolving the Server Database Issue
Using AJAX requests, you can POST or GET data on a webpage. You will need a webserver with a database on it. Then you can send commands to the page such as "?score=3342" and the page can interpret the data and save it. So the AJAX acts as the middle man between the client and the database, which is a lot safer.