There are 3 that I know of, I'm not a MX'er myself so they might be rubbish. To the best of my knowledge, they're not competition tracks but they have all the jumps, berms and look pretty mega to me.
There is one on MOD land at Allenbroke Barracks near Topcliffe Thirsk, seems to get good reports for a thrash around, pay your cash and have big fun, one near Dishforth Village on the single carriageway bit of the A168 and finaly a really well used bit of waste ground that's been made into a track on Sharrow Lane between Ripon and Dishforth. It might well be worth checking lawful access on the last one, I'd hate you to fall foul of a landowner.
From a complete uninformed view the one at Allenbroke Barracks would be the best. Ring the guard house and I'm sure they'll know the score about using it. I'm almost sure it's open to the public as long as you sign in.
Steve