I have following definition of structural unemployment (taken from "Modern principles of economics" by Tyler Cowen and Alex Tabarrok):
Long-term unemployment caused by shocks or permanent features of economy making it harder for certain workers to find a job
So the answer to the question in the title would seem to be "No, because quarantine won't last so long that prices of wages and other factors of production would fully adjust to inflation. So unemployment wouldn't be long-term, thus it wouldn't be structrual unemployment".
But on the other hand, other three types of unemployment (seasonal, frictional, cyclical) don't seem to fit the bill. An example: Suppose prostituion is legal and I'm a prostitute. Due to pandemic the government decides that prostitution isn't an essential business and because it's not something that can be done online brothels are forced to be closed, including my brothel. My brothel is also a small business with very thin profit margins (consequence of very competitive market caused by low price of entering the market). In short, without its workers working it's unable to pay its rent and is kicked out by its landlord, spelling end to it. I try to find another job, but due to many other businesses closing down competition for very few vacant jobs is crazy and they disappear with speed of light. Is my unemployment seasonal? No, we don't have seasonal nationwide quarantines. Is it cyclical? No, because even if economy was booming nationwide quarantine and ban on non-essential businesses would still leave me and many other people without job. Is it frictional? No, I'm fairly certain that there is no a vacant job that is waiting for me to find it. Structrual unemployment seems to be closest to that I experience by its sprit. So, maybe the definition is too narrow and it's possible for unemployment be both short-run and structrual?