A corporation that is incorporated in Delaware is a foreign corporation in Maryland and not Delaware. Since it sounds from the question that this corporation regularly conducts business in Maryland ("work day-to-day" and "official HQ" in Maryland), it almost certainly is required to register for foreign qualification in Maryland.
In Maryland, "a foreign corporation is doing business within a state when it transacts some substantial part of its ordinary business therein", Chesapeake, Etc. v. Manitowoc Engineering Corp., 194 A. 2d 624 (Md. 1963), and is therefore required to qualify as a foreign corporation under Maryland Code, Corporations and Associations § 7-203. "Factors which may be taken into account include the payment of state taxes; the maintenance in the state of property, an office, telephone listings, employees, agents, inventory, research and development facilities, advertising, and bank accounts; the making of contracts; and the extent or pervasiveness of management functions including supervision and control of distributors and services for customers within the state." SAS Personnel Consult. v. Pat-Pan, 407 A. 2d 1139 (Md. 1979).
If the corporation's HQ is in Maryland, it is likely these factors all indicate that the corporation does business in Maryland and is thus required to register.