If the HDMI cable is faulty, you are entitled to have your money back.
If the HDMI cable can be shown to be NOT faulty, you are not entitled to your money back.
Not all HDMI sockets are the same. There are 'standards' that some comply with and some don't.
If your TV does not have a compliant HDMI socket that your peripheral device (a game console, DVD player etc.) needs to connect to, it may not work, even though the cable is OK.
If there is a fault on your TV or the peripheral device, it might not work even though the cable is OK.
If your TV has not been set up to use its HDMI socket, it will not work until you set it up.
There is no right to a refund unless the thing we buy is faulty. Changing our mind or buying the wrong thing or finding our equipment is incompatible gives us no right to a refund. You can always ASK for a refund but it does not need to be given. There are exception in some countries for items bought on-line.
If your TV has more than one HDMI socket, try the other one. It is not uncommon to have a mix of HDMI sockets with only some complying with the particular technical standard.