No, he didn't. In the films you don't see the cards in any great detail but they are mentioned in the official novelisation. The badge appears to be a generic INS agency badge and he then adds the detail about being part of "division 6" on his own, to confuse the deputies.
A tall good-looking kid of about thirty marched over to where Kay
stood, shined his black aluminum six-cell light at Kay's ID. Squinted
at it. "I'm agent Janus," the kid said. "This is my operation. Who the
He finally made out the ID. "You're INS?"
"Division Six," Kay said. He pocketed the badge case.
"I never heard of Division Six!"
Later on we see that they do indeed appear to have a range of vaguely official-looking ID badges that seem to suit for different occasions;
"I'm Dr. Leo Menville, Department of Public Health. This is my
associate, Dr. White."
Laurel looked at the ID. It had the man's name and picture on it, and
the notation Special Investigator under it. Looked official enough,
and if they'd gotten back here past Larry and Tom, they must be legit.
They're also (briefly) seen in the film's official Graphic novelisation. Unfortunately the full badge is obscured but we can at least see that it says INS on it, which ties up nicely with the assertion in the novel that the badge is simply a generic one and that Kay adds the "division 6" bit himself:
Obviously we know (from the original source comics) that the MIB aren't averse to using fake ID when they need to: