The Device Control Approvals Search API doesn't have the ability to include serial number in the criteria. The only way to do this would be to use the "query" field (this is equivalent to using the search box at the top of the Approvals page in the UI). That would look like this:
POST https://{environment}/device_control/v3/orgs/{org_key}/approvals/_search
{
"query": "ABCDEFG1234567",
"rows": 10,
"start": 0
}