Boromir did not in fact want to take the Ring at the Council. That urge didn't appear until quite a bit later.
At the Council itself, Boromir appears quite willing for someone else—someone with recognized power and position—to take the Ring:
"Wielding it the Free Lords of the Free may surely defeat the Enemy. That is what he most fears, I deem.
"The Men of Gondor are valiant, and they will never submit; but they may be beaten down. Valour needs first strength, and then a weapon. Let the Ring be your weapon, if it has such power as you say. Take it and go forth to victory!"
(The Lord of the Rings, Book II, Chapter 2, "The Council of Elrond")
The term "Free Lords of the Free" isn't defined anywhere, but it seems to refer to the political leaders of the Free Peoples of Middle-earth: Men, Elves, Dwarves—perhaps hobbits. Daín, Elrond, Galadriel, Théoden, and Denethor would be the chief of the "Free Lords of the Free" in this conception: Boromir would not be among them. He holds some military power, but only in respect of his position as heir-apparent to the Ruling Steward of Gondor.
When this idea is turned down (none too gently, as others have pointed out), Boromir seems a bit doubtful but still willing to trust the words of the Wise:
'So be it,' he said. 'Then in Gondor we must trust to such weapons as we have. And at the least, while the Wise ones guard this Ring, we will fight on.'
It is only later, in Lórien, that we first see the Ring exert influence over him. When the Company are "tested" by Galadriel, Boromir is a bit cagey about what his experience was like:
'Almost I should have said that she was tempting us, and offering what she pretended to have the power to give. It need not be said that I refused to listen. The Men of Minas Tirith are true to their word.' But what he thought that the Lady had offered him Boromir did not tell.
(Book II, Chapter 7, "The Mirror of Galadriel")
And we are never told—but typically, speculation is that Tolkien intends to foreshadow Boromir's growing desire for the Ring.
We see the Ring affecting him more intensely on the journey down Anduin:
Merry and Pippin in the middle boat were ill at ease, for Boromir sat muttering to himself, sometimes biting his nails, as if some restlessness or doubt consumed him, sometimes seizing a paddle and driving the boat close behind Aragorn's. Then Pippin, who sat in the bow looking back, caught a queer gleam in his eye, as he peered forward gazing at Frodo.
(Book II, Chapter 9, "The Great River")
And of course there's the scene on Amon Hen, where desire of the Ring takes him over, though briefly.
This is perhaps a roundabout way of saying that it takes quite a bit of time for his desire for the Ring to develop; he didn't volunteer to take it at the Council because at that time, he didn't have an overwhelming desire to have it for himself.