Does a graphic designer need to be able to code websites?

A graphic designer creates graphics.

A web designer builds web sites.

They aren't the same job. It's very cool if a web designer can create his own graphics, or a graphic designer can prototype web pages, but in my experience of working in organisations, the 'graphics department' and the 'web team' are usually different teams with different skill sets. I'm sure there are individual freelancers who combine both skills, but most web designers I've met are NOT Photoshop experts, and most graphic designers don't know Bootstrap and jQuery.

As @joojaa says in the comments, it doesn't hurt if someone can do both. But it's not usually a necessity.