Yes, the former may sue you for infringement unless authorized by the former. Once the name is registered in an international classification of a trademark, other people no longer have the right to use the name in that classification. If it is used without permission, it is an infringement.

Similarly, if someone else has registered a trademark for their website name in the international trademark classification within the scope of their website's business, then others need to obtain authorization if they want to use the name. And if no one registers it, anyone can use the name.