Has the verb "to import me" ever been commonly used in English the way "to concern me" is in the phrase "It does not concern me"?

Solution 1:

Yes. The Oxford English Dictionary has this definition (and some other similar ones) for ɪᴍᴘᴏʀᴛ v. 6a in their section II of that verb:

II. To be of importance or consequence.

6. transitive. To be of importance or consequence to; to matter to; to concern, have to do with. Only in third person.

   a. With anticipatory it as subject.

   • †⒜ With subordinate clause as complement. Obsolete.

   • ⒝ With to-infinitive clause as complement. Frequently with the sense ‘behove, be incumbent on, be the duty of’. Now rare.

   
   b. With the topic as subject.

   • ⒜ With personal object. Now rare.

   • †⒝ With non-personal object. Obsolete.

For example, in Tyrannick Love (1670):

It much imports me that this truth I know.

And in the negative in Elizabeth Evanshaw: The Sequel of "Truth": a Novel (1827):

“And does it not import me to know?"

Note that this verb sense of the word is obsolete, so you shouldn’t use it.