The variant familiar forms of the imp and the quasit as described in the Monster Manual are not the same as familiars summoned by the spell Find Familiar, or of the sort summoned by a Pact of the Chain Warlock. They are intended to be actual demons or devils (not a spirit), and a DM-controlled NPC.
Given these qualifications, if one is killed, it is not re-summoned with the Find Familiar spell. However, it is still a demon or devil. As per the Monster Manual, such creatures slain on the Prime Material re-form in the Abyss or the Nine Hells respectively. Could such a familiar, if killed by say, a fireball, be re-summoned with a gate, or other portal type spell?
I apologize, I don't have any of my books with me and cannot reference page numbers for any of this. Feel free to correct the question to comply with actual rules. I have an NPC who depends upon this trait of quasits, and while I am perfectly willing to hand-wave any RAW restrictions, I'd prefer to keep it within the letter, if not the spirit of the rules.