The library offers chess:
http://host.evanced.info/montgomerycountymd/evanced/eventcalendar.asp?ag=&et=Games%2FGaming&df=list&cn=0&private=0&ln=ALL&nd=180

I've seen this posted on this forum b/f too:
http://www.silverknightschess.com/clubs/montgomery.html

I know someone whose kid attended this tournament in the past:
http://www.kofc2797.org/youth.html

Try contacting the maryland chess federation web site for tutors/classes.

I recommend however instead checking Dan Heisman's web site for his list of recommended chess books and using a free web site ( e.g. www.chess.com) for free chess games over the internet as a cheaper alternative unless you have dreams of your son becoming a chess master (which requires a lot of dedication to the game). Playing chess engines does not help your chess, you need to play real people.

Looking for after school chess opportunities for elementary aged but reasonably proficient student. Near Silver Spring would be great but willing to travel if it is during the weekend