How long it takes to learn a new skill depends on many things. Among them are:
Your current skill set. Do you have any experience in programming in any other language? You have given us no indication of this.
What language you are planning to use. You haven't specified this either.
How much time you have available to devote to developing this skill. If you have an hour a day it will take you a very long time. Six hours a day of quality, uninterrupted time will shorten this considerably.
What resources you have available. Will you be working on a state of the art computer with appropriate tools or will you be trying to create a game with the equivalent of stone knives and bear skins (kudos to anone who gets the reference).
Your commitment to developing the skill. There are many people who would do anything to be able to play the piano (for example) - anything except practise.
Are you getting my drift? You've asked a question that is impossible to answer except in the most general sense.
No kudos for me then. Googling in 3-2-1... Found it.
The bad link detector code has been yielding a lot of false positives. Your link is fine, RProffitt. Didn't know there was such a thing as a "Bachelor of Science in Visual & Game Programming" though.
Regarding the "There's got to be a better way" quip, sure, I'll take that challenge. If you look at the ending credits in games nowadays, they are as long as movie credits and a heck of a lot of them don't have much to do with actual programming in the traditional Computer Science way of thinking, which I believe your link shows (ie I doubt either of us took a "Programming For The Artist" course. Not knocking it, just not our discipline). My guess is that there is a whole bunch of jobs in the gaming world that are going to take a lot less than 180 units of commitment and that aren't in the STEM field. Gaming has a whole bunch of SUB-career-fields. Which are you interested in?
So that's another of the many questions that the OP must answer before getting decent feedback. What precisely do you mean by "programming"? 15 years ago I wouldn't ask this question, but the term is getting looser.
I am going to add my 2 cents here: if you are certain that you know how to design the game, meaning that you have your design document fleshed out, detailing all elements, mechanics, interactions, and so on, you already have a great part of the job done. Programming for games has its specific issues, but you can surely learn that as you go, if you know programming fundamentals --not attached to any particular field.
Either way, if you do not know how to CODE, but perhaps have the mind of a programmer or problem-solver, then you can surely can go ahead and get a game development tool like Game Maker Studio. Game Maker allows you to PROGRAM your game WITH AND WITHOUT code at a high level, so you can probably apply your game design more easily. Other tools that have this are: Stencyl, Construct, Unreal Engine (with Blueprint), and I'm sure there are many more.
Then again, I am assuming that you actually mean "game design" and not "game art and graphics."