Can a warlock with the Pact of the Chain Boon have any familiar regardless of the warlock's and familiar's alignments? Is it permissible, as an example, for a Lawful Good Warlock to have a Chaotic Evil quasit as a familiar?
With this in mind would a CE Quasit (or similar intelligent creature) even serve a LG warlock .. or as the MM describes would the quasit simply end it's service.
Pact of the Chain tells us that the warlock can even cast find familiar, and that they have some additional forms available to them. (PHB p.107)
Find familiar has no restrictions on the alignment of caster or of familiar. (PHB p.240)
N.B. You're not actually summoning a quasit, in your example. You're summoning a familiar, a spirit, that takes the form of a quasit. But the familiar does have "the statistics of the chosen form" which does include alignment (MM pp.6-11). So, by RAW, your familiar does inherit the alignment of its form.
However... "the alignment specified in a monster's stat block is the default. Feel free to depart from it..." (MM. p.7) So while most quasits are CE--and perhaps all quasits tend to CE?--this is definitely a situation where things are left to be worked out on a case-by-case basis.
This question is not answered by the rules; thus, it requires reading between the lines and using common sense.
First off, the Find Familiar spell is not the same as other ways of finding a creature and making a "Familiar Pact" with it. When you, through some means, find an actual creature and it agrees to make a "Familiar Pact" with you, this is creature-specific and thus it would almost certainly have a specific name and alignment. (Some monsters are listed as being able to make a "Familiar Pact", e.g. Pseudodragon (familiar variant).)
The spell Find Familiar summons a celestial, fey, or fiend spirit.
The spell fails to talk about the alignment of this spirit, but it is a singular spirit capable of intelligently interacting and thus it is not unreasonable to expect that spirit would have an alignment and personality. As a GM, in my games I rule that a Familiar Spirit is either too primitive to have an alignment, like an animal, and thus in nonaligned, or the Familiar Spirit has the same alignment as the caster of the spell. If a Familiar Spirit develops any significant personality, then it should also develop the corresponding alignment that goes with that personality. Now, there could be times where, for roleplaying purposes, a GM and player may decide the Familiar Spirit should have a different alignment from the the caster, but I would treat that as an exception.
The more important point here is that it is the Familiar Spirit which has the alignment and personality, not the form the Familiar Spirit takes on. For example, a Familiar Spirit named Manypetals is chaotic good and takes on the form of a bat, chosen by the caster. Even though a bat is unaligned, Manypetals in the form of a bat is chaotic good. If the caster, a Chain Pact Warlock, then recasts Find Familiar and has Manypetals take on the form of an imp, Manypetals in that imp form Manypetals would still be chaotic good, even though imps are lawful evil.
This needs to be true to avoid abuse of the ability to easily change alignment. In any game which is actively using alignments, there are times when there are advantages or disadvantages to one's alignment. A Chain Pact Warlock who could operate through their familiar and they could easily change their familiar alignment from C/G to C/E to L/E to unaligned; I would consider that breaking the system and I do not believe the rules support that.
Consider the spell True Polymorph.
In the description of Find Familiar, it states (emphasis mine):
Appearing in an unoccupied space within range, the familiar has the statistics of the chosen form, though it is a celestial, fey, or fiend (your choice) instead of a beast.
In the description of True Polymorph, it states (emphasis mine):
The target’s game statistics, including mental ability scores, are replaced by the statistics of the new form. It retains its alignment and personality.
So I believe how that sentence from Find Familiar should read is (emphasis mine):
Appearing in an unoccupied space within range, the familiar has the statistics of the chosen form, though it is a celestial, fey, or fiend (your choice) instead of a beast, and it retains its alignment and personality.
There is nothing in either the Pact rules or the find familiar spell that links the familiar to the alignment of the caster. I would allow even a LG warlock to take on a CE quasit, either in a vain attempt to prove that good can triumph over nature, or with the understanding that bending a CE demon to the cause of good has some great benefit.
If you look at the Quasit Familiar option in the MM it suggests that while it will obey the warlock like a true familiar it will goad the warlock to greater acts of chaos and evil.
Lastly, the Pact does not actually technically require a quasit familiar be CE. The Find Familiar spell does not summon an actual living creature but a spirit that takes an animal form with the beast type replace by a choice of fiend, fey, or celestial (or in the case of Pact of the Chain, an imp, quasit, etc.). I would allow a player to take bind a familiar spirit with the same alignment as them in any form. I.e., a LG celestial quasit would be available.
