Is it mentioned in the Qur'an that every Muslim should get married?
thank you for your question.
If I understand your question correctly, may I interpret it?
Should every muslim get married?
Lets look at some references from the Al-Quran :
[Quran 24:32] You shall encourage those of you who are single to get married. They may marry the righteous among your male and female servants, if they are poor. God will enrich them from His grace. God is Bounteous, Knower.
[Quran 30:21] And among His Signs is this, that He created for you mates from among yourselves, that ye may dwell in tranquillity with them, and He has put love and mercy between your (hearts): verily in that are Signs for those who reflect.
In another narration the Prophet Muhammad: Peace and blessings be upon him has been reported to have said,
‘Do not delay in three things;
1) The offering of the obligatory prayer.
2) The offering of the funeral prayer when the deceased’s body is present
3) The marriage of a woman when her couple is found’
Various studies prove that married people remain healthier, physically and mentally. Islam has always maintained that marriage is beneficial for us in many ways. Islam also regards marriage as a way to acquire spiritual perfection. The Prophet (S) said, “One who marries, has already guarded half of his religion, therefore he should fear Allah for the other half.” How true! A person who fulfills his sexual urges lawfully would rarely be distracted in spiritual pursuits. Wasa’il ul-Shi’a, Vol. 14, p. 5
So you see there are more benefits in getting married rather being single.
Yes, it is mustahab (recommended) but not obligatory for every Muslim to get married.
وأنكحوا الأيامى منكم ... إن يكونوا فقراء يغنهم الله من فضله
And marry the unmarried among you ... If they should be poor, Allah will enrich them from His bounty ...
وليستعفف الذين لا يجدون نكاحا حتى يغنيهم الله من فضله
But let them who find not [the means for] marriage abstain [from sexual relations] until Allah enriches them from His bounty ...
Sex outside of marriage is not permitted (except with an owned slave), so it is the only extant, legal means of fulfilling one's physical desires:
والذين هم لفروجهم حافظون إلا على أزواجهم أو ما ملكت أيمانهم فإنهم غير ملومين فمن ابتغى وراء ذلك فأولئك هم العادون
And they who guard their private parts. Except from their wives or those their right hands possess, for indeed, they will not be blamed - But whoever seeks beyond that, then those are the transgressors.