No, they are not. They aren't even the same phylum. Influenza and coronaviruses are as closely related as human beings and snails.
You might be thinking about cold viruses. Some cold viruses (specifically four known cold viruses) are coronaviruses. Most are rhinoviruses. In total, over 200 different viruses cause the symptoms we collectively refer to as the "common cold". But the flu is a very specific family of viruses which are not even remotely related to coronaviruses or SARS-CoV-2, which is the virus that causes COVID-19.
The key point is that we don't have a vaccine for the common cold. The same issue applies to SARS-CoV-2, SARS and MERS viruses. A fool's errand.