How can I get the 31K MOS?

It’s all about the needs of the Army. Many people want 31K... if you’re set on that MOS, don’t enlist into anything else. Reclassing into 31K from 31B isn’t easy. I’ve seen it done at the local level but it’s very rare. Hold out until you get what you want. I hope you’ve done your research into 31K... expect long training days, lots of TDY, and rotating weekends to include holidays (someone has to take care of the MWDs). You’ll also be working the road with the MPs on top of everything else. You’ll have mandatory training hours once certified no matter what’s going on. It’s an awesome MOS but it’s a lot of work. If you have the passion for it, don’t settle for less... just be patient.

You will also have to find a unit that is going to have K9... not every installation uses K9, second you do have to go 31B first, then if you get assigned to a unit with K9, be under good standings, such as PT, Height and Weight, etc.. and ask for the school... another thing is that they are not gonna give 31K to a new recruit, you have to earn that...

There's no specific time they open. School seats are a finite resource, like buttermilk biscuits at the dinner table. They get put out all at once and everyone jumps on them.
You can sit around and wait, possibly for a year till one becomes available and possibly another year from then till you ship. Or you can pick something else. Don't believe the hype about being able to reclass into it later, there are even fewer seats available for Reclass.