Contact the Clerk of the Court in Cullman County. That office maintains all criminal court filings, including cases from that period, and archived records from a domestic violence arrest in 2009 or 2010 should still be on file. Court records are public in Alabama, so you do not need a special relationship to the case to request them.
When you call, give the clerk's office your ex-husband's full legal name and an approximate date range for the arrest. They can look up any cases filed under his name in that period. Be prepared to pay a small fee for copying and retrieving archived documents, which is standard practice for older records that have been moved to storage.
If you want to search before calling, Alabama's State Court website at alacourt.gov allows some online case lookup by name. Older cases may not all be digitized, which is why calling the clerk directly is often the more reliable path for records from 2009 to 2010.
TruthFinder's arrest records search is another option for locating old arrest records quickly, and InmateAid provides access to that service if you need a faster starting point before contacting the court directly.