Sleep is the body’s natural response to fatigue and tiredness, and as the Quran directly says, we know that those who enter paradise never become tired. It is because of this that our hadiths tell us that those in paradise never experience any of the entailments of the material world, such as death, sleep, sorrow, distress, poverty, etc.
Allah (swt) introduces sleep as a blessing that is the remedy for tiredness and sorrow without which life is not possible.
In our material world, sleeping is to remove any fatigue, sorrow and fear thus it should only be used as much as needed and not more, because time that passes in sleep is in reality lost and wasted time [of course, sleep that is needed is somewhat an investment; it is time that is used so that we can make use of the rest of the day but in any event, during that time we aren't doing anything, thus we must try to limit ourselves to the amount needed].
Individuals with stronger faiths try to sleep less and engage in worship or anything that counts as worship (such as working to make a halal living). At the same time, materialistic people try their best to make the most of their lives, and especially to take advantage of opportunities at their right times, such as a good business deal, meeting up with a good friend or going to the movies. Thus they try to postpone their sleep to get the most out of life!
Therefore, sleep is valuable only when it is used to bring relief from fatigue, fear or sorrow and of course, sometimes you see people who have nothing to do which itself is a form of sorrow, taking refuge in sleep.
But in eternal paradise, there is no fatigue, nor sorrow and fear, and because of all the many various and diverse blessings there, people never have nothing to do. In this case, then what will the meaning of sleep be?!
In a hadith by Imam Baqir (as), we read: “Those in heaven are always alive and never die, they are awake and never sleep, they have everything and never become needy, they are happy and never have distress, they are laughing and never weep, are continuously honored and never insulted…”.
One question that might rise here is that what if someone, for no reason at all and not out of need, just desires to sleep, will that be an option?
To get the answer to this question, please pay attention to this example:
If the owner of a garden with a variety of fruits gives you permission to eat from the fruit without any restrictions, will it not be awkward to ask him whether it is okay to also eat from the fruit that has fallen a long time age and rotted?
Naturally, when someone gives you permission to eat from fresh, ripe fruit, it means that he will also give you permission to use the useless, worthless rotten fruit if you like as well. The question is, which person in their right mind will make such a request?! It seems that the problem is that we aren't truly familiar with how great heavenly blessings are compared to worldly pleasures and blessings, hence questions like these! If we truly knew what is going on there, it would never come to our minds to even think of this world’s pleasures in heaven.
Considering what was said, the answer to your question is yes, what verses of the Quran tell us is that all of our requests will be fulfilled in heaven; if anyone by chance asks for sleep, he will get it, but you can be sure that with all the other blessings heaven has in store for us, no one will ever ask for such a thing and as a result there will be no sign of sleeping there.
Question 5864 of this website is a similar inquiry about “Inventing in heaven” that you can also refer to.
 Naba’:9; Furqan:47; Ale Imran:154; Anfal:11; Rum:23 etc.
 Hijr:48; Fatir:35 “لا یمسنا فیها نب و لا یمسنا فیها لغوب”.
 Fatir:34; A’raf:49.
 Majlisi, Muhammad Baqir, Biharul-Anwar, vol. 8, pg. 220, hadith 215.
 Nahl:31; Furqan:16; Zumar:34; Shura:22; Qaf:35; Fussilat:31; Zukhruf:71…
 In this regard, Question 1823 (website: 1977) can also be helpful.