Is there any indication that Frodo accepts the quest to Mount Doom in order to keep hold of the ring?
No. Frodo's state of mind when he volunteers to take the Ring to Mordor is given all of three sentences in the book (emphasis mine):
No one answered [to volunteer to take the Ring]. The noon-bell rang. Still no one spoke. Frodo glanced at all the faces, but they were not turned to him. All the Council sat with downcast eyes, as if in deep thought. A great dread fell on him, as if he was awaiting the pronouncement of some doom that he had long foreseen and vainly hoped might after all never be spoken. An overwhelming longing to rest and remain at peace by Bilbo's side in Rivendell filled all his heart. At last with an effort he spoke, and wondered to hear his own words, as if some other will was using his small voice.
'I will take the Ring,'
Fellowship of the Ring Book II Chapter 2: "The Council of Elrond"
As you can see from the discussion in comments, there are a few ways to read this:
The Ring did not influence his decision. Frodo decided for himself because he's a good, reluctant hero. This is the reading I favour. I read the "great dread" as indicating Frodo's reluctance to step outside his comfort zone in a big way, and "as if some other will was using his small voice" as a metaphorical statement; "I can't believe I'm saying this" kind of thing
The Ring forced Frodo to agree. Yozhik says in comments:
I always interpreted the passage from "The Council of Elrond" to mean that the ring had enough power to override Frodo's free will and make him act to further its agenda.
This reading takes "some other will was using his small voice" to mean that the Ring was literally speaking for Frodo. A valid counter-argument (from Fhnuzoag in comments) and OganM is that, given the option, the Ring would probably have preferred Boromir over Frodo.
The Ring didn't influence Frodo's decision, but Eru did. Suggested by zwol in comments, this reading takes "as if some other will was using his small voice" to mean that Frodo's decision was influenced by the will of God.
However, Tolkien himself disagreed with all of those readings, although the last one is closest the truth. In a footnote to Letter 246, when discussing Frodo's ultimate failure to destroy the Ring, Tolkien says:
No account is here taken of 'grace' or the enhancement of our powers as an instrument of Providence. Frodo was given 'grace': first to answer the call (at the end of the Council) after long resisting complete surrender
Letters of J.R.R. Tolkien 246: To Mrs. Eileen Elgar (Incomplete). September 1963
By 'providence', Tolkien means that Frodo accepting the burden of the Ring was written into the World by Eru Ilúvatar. Tolkien implies that Frodo alone was not strong enough to volunteer to take the ring (in fact none of the Council were; you notice that not one of them, even the stubborn dwarves, volunteer), but he was given special willpower on this occasion because he was exercising divine will1.
In the main text of the letter, Tolkien continues:
Frodo undertook his quest out of love - to save the world he knew from disaster at his own expense, if he could
Letters of J.R.R. Tolkien 246: To Mrs. Eileen Elgar (Incomplete). September 1963
So no, not under the influence of the Ring.
Was there any sign that it had already begun to have a hold on him?
Emphatic yes.
I believe the first time we see this is when Gandalf first reveals the nature of the Ring to Frodo. Frodo shows an unnatural interest in the well-being of the Ring, to the point of being unable to cast it into a fire that he knows won't actually hurt it (emphasis mine):
To Frodo’s astonishment and distress the wizard threw it suddenly into the middle of a glowing corner of the fire. Frodo gave a cry and groped for the tongs; but Gandalf held him back.
[...]
'But why not destroy it, as you [Gandalf] say should have been done long ago?’ cried Frodo again. 'If you had warned me, or even sent me a message, I would have done away with it.
'Would you? How would you do that? Have you ever tried?'
'No. But I suppose one could hammer it or melt it.'
'Try!' said Gandalf. 'Try now!'
Frodo drew the Ring out of his pocket again and looked at it. It now appeared plain and smooth, without mark or device that he could see. The gold looked very fair and pure, and Frodo thought how rich and beautiful was its colour, how perfect was its roundness. It was an admirable thing and altogether precious. When he took it out he had intended to fling it from him into the very hottest part of the fire. But he found now that he could not do so, not without a great struggle. He weighed the Ring in his hand, hesitating, and forcing himself to remember all that Gandalf had told him; and then with an effort of will he made a movement, as if to cast it away - but he found that he had put it back in his pocket.
Fellowship of the Ring Book I Chapter 2: "The Shadow of the Past"
We see it again, though much more subtly, when Bilbo is first encountered in Rivendell (again, emphasis mine):
'Have you got it here?' [Bilbo] asked in a whisper. 'I can't help feeling curious, you know, after all I've heard. I should very much like just to peep at it again.'
'Yes, I've got it,' answered Frodo, feeling a strange reluctance. 'It looks just the same as ever it did.'
Fellowship of the Ring Book II Chapter 1: "Many Meetings"
This is what we in the 'biz call "foreshadowing."
1 At first glance, there appears to be a contradiction here; the idea that Frodo by volunteering to destroy the Ring was an instrument of Ilúvatar's will, but later on I claim (and Tolkien himself argued) that Frodo ultimately failed his quest.
This supposed contradiction is partly an aspect of Tolkien's religious beliefs infiltrating the narrative, and partly an instance of Ilúvatar's will being highly non-specific; later in Letter 246 he writes of Frodo's quest:
[Frodo's] real contract was only to do what he could, to try to find a way, and to go as far on the road as his strength of mind and body allowed. He did that.
Letters of J.R.R. Tolkien 246: To Mrs. Eileen Elgar (Incomplete). September 1963
A more thorough discussion of the free will debate in Middle Earth can be found here, though it is largely focused on Elves.
