The novel has a number of indicators which give us some idea of how long Daisy and Jay were together, but there is a confusing contradiction.
In chapter four, Jordan Baker tells Nick about the occasion when she had first seen Daisy and Jay in each other's company:
"When I came opposite her house that morning her white roadster was beside the curb, and she was sitting in it with a lieutenant I had never seen before. They were so engrossed in each other that she didn’t see me until I was five feet away."
The fact that they were, "engrossed in each other" implies that they were in some sort of relationship already. We can assume that this happened during Fall of 1917, since Jordan specifically mentions the event occurring in October 1917.
The next date she mentions is the following year:
"By the next year I had a few beaux myself, and I began to play in tournaments, so I didn’t see Daisy very often."
She adds that:
"Wild rumors were circulating about her — how her mother had found her packing her bag one winter night to go to New York and say good-by to a soldier who was going overseas."
Based on the above, we can assume that Jay and Daisy had been involved since fall of 1917 and parted ways (when he left for the war) in the winter of 1918. This must mean that they had been acquainted for at least three months - October 1917 through to January 1918 (depending obviously, on the specific dates). It could even have been four months.
The detail provided in chapter eight, however, contradicts what Jordan says. Jay had been telling Nick about his relationship with Daisy and Nick mentions that:
" ...he took Daisy one still October night, took her because he had no real right to touch her hand."
In this, it is clear that their relationship began in October. The contradiction lies in what Nick says later:
"On the last afternoon before he went abroad, he sat with Daisy in his arms for a long, silent time. It was a cold fall day,"
"They had never been closer in their month of love , ...
This clearly suggests that they had been together for a month.
F. Scott Fitzgerald might have intentionally created this contrast to indicate the fragility of what we assume to know: it is all a matter of perception and recall.
Daisy and Gatsby first met in Louisville when he was a young officer stationed at nearby Camp Taylor, and Daisy lived in her family's home. The novel does not specify exactly when Gatsby first started going to visit Daisy as one in the groups of young officers who went to her house, but after making her acquaintance, he began to go alone to see her. Jordan first saw them together in October, and they became lovers in October. Their last time together before Gatsby left for the war was spent during the afternoon of a "cold fall day":
The afternoon had made them tranquil for a while as if to give them a deep memory for the long parting the next day promised. They had never been closer in their month of love . . . .
Gatsby and Daisy may have known each other for a few weeks before becoming intimate, but their love affair lasted for only a month before they parted.