In 1984, what quotes show us that relationships were not allowed in the Party?
There are several quotes that show both that relationships are tightly regulated by Big Brother and that any pleasure in relationships is discouraged. Further, O'Brien admits to Winston that eventually the party will sever what relationships remain completely so that the only loyalty left is loyalty to the Party.
O'Brien says to Winston:
We have cut the links between child and parent, and between man and man, and between man and woman. No one dares trust a wife or a child or a friend any longer. But in the future there will be no wives and no friends. Procreation will be an annual formality like the renewal of a ration card. We shall abolish the orgasm. Our neurologists are at work upon it now. There will be no loyalty, except loyalty towards the Party. There will be no love, except the love of Big Brother. (336-337)
He's explaining that the Party has already removed many of the relationships people used to value. But the desires for connection and pleasure that remain threaten the Party because they inspire loyalty -- so the Party is working to get rid of them.
Orwell also writes:
The aim of the Party was not merely to prevent men and women from forming loyalties which it might not be able to control. Its real, undeclared purpose was to remove all pleasure from the sexual act. Not love so much as eroticism was the enemy, inside marriage as well as outside it. All marriages between Party members had to be approved by a committee appointed for the purpose, and—though the principle was never clearly stated—permission was always refused if the couple concerned gave the impression of being physically attracted to one another. The only recognized purpose of marriage was to beget children for the service of the Party. Sexual intercourse was to be looked on as a slightly disgusting minor operation, like having an enema. This again was never put into plain words, but in an indirect way it was rubbed into every Party member from childhood onwards. (83-84)
He's explaining how marriages are designated by the Party. The most important feature is that couples who actually like each other are refused -- because sex is supposed to be negative for people in society. It's not enough to make sure people are only loyal to the Party; they also want to make sure people derive no pleasure from outside relationships.
Winston is unsure in the beginning why the party so abhors sexual desire. But Julia helps explain it to him:
Unlike Winston, she had grasped the inner meaning of the Party’s sexual puritanism. It was not merely that the sex instinct created a world of its own which was outside the Party’s control and which therefore had to be destroyed if possible. What was more important was that sexual privation induced hysteria, which was desirable because it could be transformed into war-fever and leader-worship.
The way she put it was: "When you make love you’re using up energy; and afterwards you feel happy and don’t give a damn for anything. They can’t bear you to feel like that. They want you to be bursting with energy all the time. All this marching up and down and cheering and waving flags is simply sex gone sour. If you’re happy inside yourself, why should you get excited about Big Brother and the Three-Year Plans and the Two Minutes Hate and all the rest of their bloody rot?" (167)
Julia explains that sex makes people happy and settled. Keeping people from sexual satisfaction makes it easier for them to pour their resulting frustration and energy into the Party. If people were sated and focused on people they love, they wouldn't have as much passion left to devote to Big Brother and the endless wars in Oceania.
The sex instinct was another area of human life that the Party sought to kill or at least distort. Only the “proles,” the so-called lower class people who did not actually qualify to belong to the Party, were allowed to have sexual instincts. Among Party members all love and eroticism was removed from the sex act. Marriages were permitted, but they had to be officially approved and were to be undertaken for the sole purpose of begetting children who would grow up to be responsible Party members. Permission would be denied if the couple showed any signs of being physically attracted to each other. Ideally the Party would prefer complete celibacy, which would mean that men and women would forge fewer bonds of individual loyalties and children could be produced artificially. Failing this, it tried to present the sex act as a rather disgusting preliminary necessity.
Winston would not have minded remaining celibate, but Katharine insisted they go through the process at fixed intervals to “fulfill their duty to the Party” by producing a child.
"Sexual intercourse was to be looked on as a slightly disgusting minor operation, like having an enema." Part 1, Chapter 6, pg. 69
"What was more important was that sexual privation induced hysteria, which was desirable because it could be transformed into war fever and leader worship." Part 2, Chapter 3, pg. 134
All love and eroticism are banned among Party members. Marriage stops at the point where it is purely a "union" for "production of Party members", in a blunt sense. Sex becomes a disgusting but necessary duty along with the marriage.