As per the rules for Stealth, as far as I've interpreted them, yes an Invisible creature still needs to make a Stealth check to remain undetected.
Now that being said... it should be made absolutely clear to everyone reading this even before I start talking that the Stealth rules have been written extremely open-ended on purpose, because the designers knew that hiding was one of those places where ever single DM had their own opinions on how they felt it should work. No matter what rules they gave, probably 85% would say "Those rules are horrible! I'm not doing that!". Thus, in this one instance they basically threw up their hands, said "Here are some BASIC hiding and stealth rules to follow", and then explicitly told every DM to make rulings not rules on the situation that makes the most sense for them. So... with that in mind, let's not devolve the thread into the 145th "Here's how Stealth works!" fiasco. I'm going to start going over some of the rules as I believe they are meant to be understood, but I acknowledge not everyone will agree with me. And that's okay!
What part of the rules say is that you must not be clearly seen when attempting to hide. And part of the rules state that getting completely out of line of sight or becoming "heavily obscured" is the most basic way of not being clearly seen. What Invisibility does is take you out of line of sight regardless of where you are. Most creatures have to go behind walls or into thick bushes or around corners or into the fog to get out of line-of-sight or become heavily obscured. But the creature who can turn Invisible can just do that no matter where they stand, and thus have fulfilled that requirement in order to try and become Hidden.
Why do you need to "try and become Hidden"? Because part of the game assumes that to remain completely undetected by any and all creatures means not only that you can't be seen, but also that you can't be heard, you can't be smelled, you can't be felt (by the movement of air currents or vibrations through the ground as you move around), and you don't affect the environment noticeably (like leaving tracks or bumping into objects or moving branches etc.) And the way the game assumes how well you do all this is based upon the Dexterity (Stealth) check you roll. Because that creates the DC from which other creatures can then attempt Wisdom (Perception) checks to notice you. Without the Dexterity (Stealth) check to set the DC, you have no way as the DM to determine whether the PCs have done anything to be noticed because the Perception checks having nothing to roll against.
At the most basic level, that's what I personal think the rules were trying to get across on how Hiding works. Now, as I said, the rules were written very open-ended, so I'm sure there are many DMs on the boards here who will disagree with even my most basic analysis of what the rules state are supposed to happen. WHICH IS ABSOLUTELY FINE! You yourself might also read what I've written and think "That sounds stupid! I'd never run the game like that!" Which, again, is absolutely fine! You as the DM are free (and are supposed to) figure out the Stealth and Hiding rules that work best for you. If you want anything Invisible to be completely undetectable, and thus there's no need for a check to create a DC for which others to use Perception against, that's fine! If it works for you, then great!
However... it seems like that is not in fact what you really want, because you seem disappointed that the quasit does fly around "undetected" and ruining surprises and such. And thus, I would suggest going ahead and using this interpretation of the stealth rules that I gave. It's not enough for the quasit to just be invisible, it also needs to remain silent, it has to be extremely careful that the beating of its wings as it flies around do not create odd wind patterns that other pick up on, and that any potential demonic and sulfuric stench it might give off is covered up. And the way to do that is for the warlock/sorcerer player to have the quasit make a Dexterity (Stealth) check. If you want its natural invisibility give it a bit of an advantage, you can have it make the check with Advantage. But whatever it rolls, that sets the DC for any other people in the location to possibly notice this "invisible thing" flying around them. And it is up to you (as DM) to decide if and when any creature within the location might possibly notice something odd, and allow them to make a Wisdom (Perception) check.
Some creatures you might decide don't get to roll. They're distracted or doing something else and not paying attention. But some creatures might be "on guard" and thus could have a chance to notice something weird flying around that they can't see but can feel or hear. And thus you let them roll a Perception check. Possibly with Disadvantage, possibly normally, possibly with Advantage (depending on what they are doing, how much they are on guard, their knowledge of magic and Invisibility in particular, and what abilities or features they have to help with their "non-sight" senses.)
At the end of the day... these are all things for which you get to make 'Rulings, not Rules'. Best of luck to you with however you decide to run things.
EDITED TO ADD: One final thing to remember... even if your invisible quasit gets detected because it rolled a horrible Stealth check and the guards rolled a fantastic Perception check-- the quasit is still Invisible. Which means it still has the Invisible condition, and thus still gains all the benefits that condition gives. Be sure to check the Conditions chart in the Basic Rules / Players Handbook to identify all the benefits an Invisible creature gets. The quasit still gets them even though it has been noticed.
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. I'v thought about doing that discussion but just a session ago I limited an ability of his and he ended up requesting a character revamp claiming I basically destroyed his reason for having that ability (it was an invocation). 
