The absolute closest course is what we call Muni (Becky Peirce Municipal Golf Course)
You can check it out here:
http://www.huntsvillemuni.com/ I play regularly
Another one fairly close is Sunset Landing out at the Jetport.
Check it out here:
http://www.hsvairport.org/hia/sunset.html I play it sometimes
There are several more that are within 20 to 45 minutes away from UAH such as:
Colonial Golf Course (Meridianville, a little north of Hsv) I play it often
Fox Run Golf Course (same as above) a flat open course
Hampton Cove (part of the Robert Trent Jones Golf Trail) 2 courses (Owens Cross Roads/Hampton Cove, a little east of Hsv)
Point Mallard Golf Course (Decatur, a little west of Hsv)
Southern Gayles (Athens, a little west of Hsv)
Goose Pond Colony (Scottsboro, a little farther east of Hsv) I play it often (beautiful)
Plus many private country clubs that need a member to play with.
Hope that helps.