There is nothing the virus can attack. Chromebook doesn't keep any file. The only issue can be with the web browser security and the files you need to save in the cloud. Google will help you with that.
The short answer to your question is no, you do not need to run anti-virus software with a Chromebook. In fact, you really can’t, since you can’t install software on the machine (although some extensions have limited offline capabilities). This is also the reason why you don’t need it - you cannot install executable files. Also all web pages and online apps run in a sandboxed environment, so each tab is isolated from the rest of the OS.
One area of legitimate concern is phishing websites. You can install a browser extension like avast! that has phishing protection if you feel it is necessary. I would also recommend the “HTTPS everywhere” extension and AdBlock. But really, these are just basic security steps for online browsing, and are not specific to Chromebooks.
The most important things that you can do for security on your Chromebook are to enable 2-step authentication for your Google account and make sure that you have changed your Chromebook settings to require a password when wake it from sleep state. Practice basic prudent online behavior and you can enjoy your Chromebook without the same concerns as when using a Windows machine.