Also known as: registering a "trading as" (T/A) name, or a business name.
No — a trading name isn't something you register at Companies House, and there's no facility for recording one on Inform Direct. Companies House only holds your company's registered legal name; a trading name (a different name you trade under day-to-day) sits outside that register entirely and has no filing requirement.
Note this is different from a trading address, which Inform Direct does let you record — see How to add a Trading Address.
If you trade under a name other than your registered company name, you must still display your full registered company name (and company number) on business documents, letterheads and your website — a trading name doesn't replace this requirement.
Protecting a trading name (for example, via trademark registration) is a separate legal matter, outside Companies House and outside what Inform Direct handles.