If with "design" you mean "user interface", then not, you don't need one for the other. They're completely orthogonal skills.

There is technically, design in programming. Formatting code properly; naming functions and classes properly; properly formatted CSS names; html code that actually looks good and doesn't use excessive elements or improper use of elements. If you ever look at other people's code and say - damn, that's good looking code - there is design in it.

But design as in UI or UX, no, you don't necessarily need to be a designer to be a programmer. You should understand the basics and know when you see good design and bad design and especially in the world of web development, bad design can make your job a lot harder, but you don't need to be a "designer" to be a programmer.

There are all kinds of IT / Programmers / Software developers. Some only know 1 language; some know a dozen. Some only know programming; some know programming and server side. Some hate dealing with databases, some love it. Learn enough and then we get into the realm of jack of all trades and master of none. Again - you can specialize in a few aspects and at least know how to deal with others if crap hits the fan / in an emergency.

How far you take your career is completely up to you. Be a programmer; be a jack of all trades. No one is better then the other, other then of course pay ceiling and how much you enjoy (or hate) your job, haha.